MODULUS OF ELASTICITY BASED METHOD FOR

ESTIMATING THE VERTICAL MOVEMENT OF

NATURAL UNSATURATED EXPANSIVE SOILS

Hana Hussin Adem

Thesis submitted to the Faculty of Graduate and Postdoctoral Studies

in partial fulfillment of the requirements for the Doctor in Philosophy degree in Civil Engineering

Department of Civil Engineering Faculty of Engineering University of Ottawa

Ottawa, Ontario, Canada

© Hana Hussin Adem, Ottawa, Canada, 2015

ABSTRACT

Expansive soils are widely distributed in arid and semi-arid regions around the world and

are typically found in a state of unsaturated condition. These soils are constituted of the

clay mineral montmorillonite that is highly active and contributes significantly to volume

changes of soils due to variations in the natural water content conditions. The volume

changes of expansive soils often cause damage to lightly loaded structures. The costs

associated with the damage to lightly loaded structures constructed on expansive soils in

the United States alone were estimated as $2.3 billion per year in 1973, which increased

to $13 billion per year by 2009. In other words, these damages have increased more

than five fold during the last four decades. Similar trends in damages were also reported

in other countries (e.g., Australia, China, France, Saudi Arabia, United Kingdom, etc.).

Numerous methods have been proposed in the literature over the past 50 years for the

prediction of the volume change movement of expansive soils. However, the focus of

these methods has been towards estimating the maximum potential heave, which occurs

when soils attain the saturation condition. The results of heave estimation considering

saturated soil conditions are not always useful in engineering practice. This is because

most of damages due to expansive soils often occur prior to reaching the saturation

condition. A reliable design of structures on expansive soils is likely if the anticipated

soil movements in the field can be reliably estimated over time, taking into account the

influence of environmental factors. Limited studies are reported in the literature during

the past decade in this direction to estimate/predict the expansive soil movements over

time. The existing methods, however, suffer from the need to run expensive and time

consuming tests. In addition, verification of these studies for different natural expansive

soils has been rather limited.

A simple approach, which is referred to as a modulus of elasticity based method

(MEBM), is proposed in this study for the prediction of the heave/shrinkage movements

of natural expansive soils over time. The proposed MEBM is based on a simplified

constitutive relationship used for the first time to estimate the vertical soil movements

with respect to time in terms of the matric suction variations and the corresponding

ii

values of the modulus of elasticity. The finite element program VADOSE/W (Geo-Slope

2007) for simulating the soil-atmospheric interactions is used as a tool to estimate the

changes in matric suction over time. A semi-empirical model that was originally

proposed by Vanapalli and Oh (2010) for fine-grained soils has been investigated and

extended for unsaturated expansive soils to estimate the variation of the modulus of

elasticity with respect to matric suction in the constitutive relationship of the proposed

method. The MEBM has been tested for its validity in five case studies from the

literature for a wide variety of site and environmental conditions, from Canada, China,

and the United States. For each case study, factors influencing the volume change

behavior of soils, such as climate conditions, soil cracks, lawn irrigation, and cover

type (pavement, vegetation), are successfully modeled over the period of each

simulation. The proposed MEBM provides good predictions of soil movements with

respect to time for all the case studies. The MEBM is simple and efficient for the

prediction of vertical movements of natural expansive soils underlying lightly loaded

structures.

In addition, a new dimensionless model is also proposed, based on the dimensional

analysis approach, for the estimation of the modulus of elasticity which can also be used

in the constitutive relationship of the MEBM. The dimensional model is rigorous and

takes into account the most significant influencing parameters such as matric suction, net

confining stress, initial void ratio, and degree of saturation. This model provides a

comprehensive characterization of the modulus of elasticity of expansive soils under

unsaturated conditions for different scenarios of loading conditions (i.e., both lightly and

heavily loaded structures).

The results of the present study are encouraging for proposing guidelines based on further

investigations and research studies for the rational design of pavements, shallow and deep

foundations placed on/in expansive soils using the mechanics of unsaturated soils.

iii

DEDICATION

This thesis is dedicated to my beloved family members, my father (Hussin El-Saete) and

my brother-in-law (Sami El-Selene), who were looking forward to seeing this

accomplishment possible!

iv

ACKNOWLEDGMENTS

I would like to thank first and foremost my supervisor, Dr. Sai Vanapalli, for his meticulous

guidance throughout the course of my stay at the University of Ottawa. I greatly appreciate

his advice, encouragement, and supervision which made the completion of this dissertation

possible. My gratitude is also extended to the committee members of my Ph.D. defense, Dr.

Braja Das, Dr. Mohammad Rayhani, Dr. Ioan Nistor, and Dr. Mamadou Fall, for their time

and valuable comments.

I thank the Ministry of Higher Education and Scientific Research – Libya for granting me the

Libyan-North American Scholarship which guaranteed a smooth completeness of my thesis.

Special thanks go to the Canadian Bureau for International Education, CBIE in Ottawa,

Canada for managing the scholarship program.

I also appreciate the help received from Department of Civil Engineering staff at the

University of Ottawa. I am also indebted to all my friends and my colleagues who were part

of my graduate life at the University of Ottawa. Uncountable help received from my

roommate and my friend Dr. Amina Mohammed is gratefully acknowledged.

Words can fail in expressing my love and gratitude to my family. Through turbulent times

and calm, through creative periods and fallow, there is/was always family who provided me

with the required encouragement and moral support.

v

LIST OF CONTENTS

Abstract ........................................................................................................... ii

Dedication ........................................................................................................ iv

Acknowledgments ........................................................................................... v

List of Contents ............................................................................................... vi

List of Figures ................................................................................................. xi

List of Tables ................................................................................................... xviii

List of Symbols ................................................................................................ xx

CHAPTER 1. Introduction ............................................................................. 1

1.1 Background ........................................................................................ 1

1.2 Objective ............................................................................................. 4

1.3 Research Methodology ....................................................................... 5

1.4 Novelty of the Research Study ........................................................... 7

1.5 Layout of the Thesis ........................................................................... 8

CHAPTER 2. Literature Review ........................................................................... 11

2.1 Introduction ........................................................................................ 11

2.2 Expansive Soil Mineralogy ................................................................. 11

2.3 Mechanism of Soil Swelling ................................................................ 15

2.4 Volume Change Movement of Expansive Soil ................................... 16

2.5 Factors Affecting Soil Volume Change ............................................. 18

2.6 Soil Suction in Unsaturated Soils ...................................................... 20

vi

2.6.1 Matric suction ....................................................................................... 21 2.6.2 Osmotic suction .................................................................................... 23 2.6.3 Matric suction profile ........................................................................... 25

2.7 Modeling of Unsaturated Flow and Atmospheric Interactions ........ 26

2.8 Volume Change Theory of Unsaturated Soils ................................... 29 2.8.1 Stress state variables ..................................................................... 29 2.8.2 Volume change constitutive relationships ..................................... 34 2.8.3 Coupled consolidation theory for unsaturated soils ....................... 42

2.8.3.1 Equilibrium equations for soil structure ........................................ 42 2.8.3.2 Water continuity equation ............................................................. 43

2.9 Volume Change Predictions .............................................................. 45 2.9.1 Methods for predicting heave potential ......................................... 45

2.9.1.1 Oedometer methods ................................................................ 46 2.9.1.2 Empirical methods .................................................................. 55 2.9.1.3 Suction-based methods ........................................................... 58

2.9.2 Methods for predicting soil vertical movement over time .............. 65 2.9.2.1 Consolidation theory-based methods ...................................... 66 2.9.2.2 Water content-based methods ................................................. 72 2.9.2.3 Suction-based methods ........................................................... 76

2.10 Summary .......................................................................................... 79

CHAPTER 3. Modulus of Elasticity of Unsaturated Expansive Soils ................ 83

3.1 Introduction ..................................................................................... 83

3.2 Background ...................................................................................... 84

3.3 Triaxial Shear Test Results and Soils Properties ........................... 90

3.4 Analysis of the Triaxial Tests Results ............................................. 98

3.5 Comparison between the Experimental and Predicted Values of the Modulus of Elasticity .................................................................

103

3.6 The Relationship between the Plasticity Index IP, the Net Confining Stress 3( )auσ − , and the Fitting Parameter α .....................................

107

3.7 Summary .......................................................................................... 110

CHAPTER 4. Proposed Approach for Predicting Vertical Movements of Expansive Soils .........................................................................................................

111

4.1 Introduction ..................................................................................... 111

vii

4.2 Constitutive Relationship for Estimating Expansive Soil Movements over Time .....................................................................

114

4.3 Key Parameters for Predicting the Expansive Soil Movements .... 118 4.3.1 Matric suction variations ............................................................... 118 4.3.2 Soil modulus of elasticity associated with matric suction ............. 120

4.4 Step-by-Step Procedure of the Proposed MEBM Approach ......... 121

4.5 Summary .......................................................................................... 123

CHAPTER 5. Validation of the Proposed Modulus of Elasticity Based Method ......................................................................................................................

125

5.1 Introduction ..................................................................................... 125

5.2 Case Studies ..................................................................................... 126

5.3 Case Study A: a Slab-on-Ground Placed on Regina Expansive Clay Subjected to a Constant Infiltration Rate (Vu and Fredlund 2006) .................................................................................................

127 5.3.1 Simulation of matric suction changes over time .................................. 129 5.3.2 Estimation of soil modulus of elasticity associated with matric

suction ...................................................................................................

131 5.3.3 Prediction of soil heave over time......................................................... 135 5.3.4 Analysis and discussion ........................................................................ 136

5.4 Case Study B: a Light Industrial Building in North-Central Regina, Saskatchewan, Canada (Yoshida et al. 1983, Vu and Fredlund 2004) .................................................................................

137 5.4.1 Simulation of matric suction changes over time ................................... 139 5.4.2 Estimation of soil modulus of elasticity associated with matric

suction ...................................................................................................

141 5.4.3 Prediction of soil heave over time ........................................................ 142 5.4.4 Analysis and discussion ........................................................................ 143

5.5 Case Study C: a Test Site in Regina, Saskatchewan, Canada (Ito and Hu 2011) ....................................................................................

144

5.5.1 Site description ..................................................................................... 145 5.5.2 Simulation of matric suction changes over time .................................. 150 5.5.3 Estimation of soil modulus of elasticity associated with matic

suction....................................................................................................

153 5.5.4 Prediction of vertical soil movement over time .................................... 153 5.5.5 Analysis and discussion ........................................................................ 155

5.6 Case Study D: a Cut-Slope in an Expansive Soil in Zao-Yang, Hubie, China (Ng et al. 2003) ..........................................................

158

5.6.1 Description of the field study ............................................................... 158

viii

5.6.2 Simulation of matric suction changes over time .................................. 161 5.6.2.1 Model calibration ........................................................................... 163 5.6.2.2 Model validation ............................................................................ 166 5.6.2.3 Simulation of matric suction fluctuation .................................... 168

5.6.3 Estimation of soil modulus of elasticity associated with matric suction ...................................................................................................

169

5.6.4 Prediction of vertical soil movement over time .................................... 169

5.7 Case Study E: a Field Site in Arlington, Texas, USA (Briaud et al. 2003) ..................................................................................

172

5.7.1 Site description ..................................................................................... 173 5.7.2 Simulation of matric suction changes over time .................................. 176 5.7.3 Estimation of soil modulus of elasticity associated with matric

suction ...................................................................................................

180 5.7.4 Prediction of vertical soil movement over time .................................... 182

5.8 Summary .................................................................................................. 185

CHAPTER 6. Elasticity Moduli of Unsaturated Expansive Soils from Dimensional Analysis ...............................................................................................

186

6.1 Introduction ............................................................................................. 186

6.2 Dimensional Analysis Background ........................................................ 188

6.3 Dimensional Analysis and Combination of Parameters ...................... 189

6.4 Triaxial Tests Results Used in the Dimensional Analysis .................... 192

6.5 The Application of Dimensional Analysis for Estimating the Soil Modulus of Elasticity .................................................................

193

6.6 Verification of the Proposed Dimensionless Model............................... 200

6.7 Prediction of Vertical Movement of Unsaturated Expansive Soils Based on the Dimensionless Model ...............................................

202

6.8 Summary .................................................................................................. 205

CHAPTER 7. Conclusions and Future Research Suggestions ............................ 206

7.1 Introduction ............................................................................................. 206

7.2 Conclusions .............................................................................................. 207 7.2.1 Overall performance of the MEBM approach ...................................... 207 7.2.2 Soil-atmospheric interaction ................................................................. 209 7.2.3 Unsaturated modulus of elasticity ........................................................ 210

ix

7.3 Recommendations and Suggestions for Future Research Studies....... 211

REFERENCES ................................................................................................ 213

x

LIST OF FIGURES

1.1 The annual costs of damage to structures constructed on expansive soils in the United States since 1973.......................................................................................

3

2.1 A single silica tetrahedron and the sheet structure of silica tetrahedrons arranged in a hexagonal network (Mitchell and Soga 2005).................................

13

2.2 A single octahedral unit and the sheet structure of the octahedral units (Mitchell and Soga 2005)......................................................................................

13

2.3 Schematic diagrams of the structures of (a) kaolinite, (b) montmorillonite, (c) illite (Mitchell and Soga 2005)..............................................................................

14

2.4 The intercalation of water molecules in the inter-plane space of montmorillonite (Taboada 2003)...........................................................................

15

2.5 The orientation of water films around high charge density clays (Mitchell and Soga 2005)...............................................................................................

16

2.6 Type of expansive soil movements: (a) soil volumetric expansion in three directions when there is no restriction, (b) soil heave when lateral movement is restricted................................................................................................................

16

2.7 Vertical movement of expansive foundation soils: (a) edge uplift, (b) doming heave (modified after Department of the Army USA 1983).................................

18

2.8 The capillary phenomenon contributing to the matric suction (Mitchell and Soga 2005).............................................................................................................

22

2.9 Relationship among pore radius, matric suction, and capillary height (Fredlund and Rahardjo 1993)...............................................................................................

22

2.10 Microscopic water-soil interaction in unsaturated soils: (a) negative pore pressure acts all around the particles, (b) suction forces act only at particles contact (Mitchell and Soga 2005)..........................................................................

23

2.11 Typical pore water pressure profiles (modified after Fredlund and Rahardjo 1993)......................................................................................................................

25

2.12 Hydraulic conductivity as a function of soil suction (Benson 2007).................... 27

2.13 Soil-water characteristic curve and specific water capacity (Benson 2007)......... 28

xi

2.14 Three-dimensional constitutive surfaces for unsaturated soil: (a) soil structure constitutive surface, (b) water phase constitutive surface (modified after Fredlund et al. 2012)..............................................................................................

39

2.15 Three-dimensional constitutive surfaces for unsaturated soil expressed using soil mechanics terminology: (a) void ratio constitutive surface, (b) water content constitutive surface (modified after Fredlund et al. 2012).......................

41

2.16 Stress path followed when using the direct method (Fredlund et al. 1980).......... 47

2.17 Heave calculations using the direct method (NAVFAC 1971).............................. 48

2.18 Stress path followed when using Sullivan and McClelland method (Fredlund et al. 1980).................................................................................................................

50

2.19 Adjustment of laboratory test data to compensate for compressibility of oedometer apparatus (Fredlund and Rahardjo 1993)............................................

50

2.20 Construction procedure to correct for sampling disturbance (Fredlund and Rahardjo 1993)......................................................................................................

51

2.21 Stress paths followed when using double oedometer method (Jennings and Knight 1957)..........................................................................................................

52

2.22 Hypothetical oedometer test results (modified after Nelson and Miller 1992)..... 54

2.23 Idealized void ratio versus logarithm of suction relationship for a representative sample (modified after Hamberg 1985)................................................................

59

2.24 Measured and predicted heaves with depth under the center of the slab (modified after Vu and Fredlund 2004).................................................................

67

2.25 Measured and predicted heaves at the surface of the slab (modified after Vu and Fredlund 2004)................................................................................................

67

2.26 Soil movement predicted by Zhang (2004) method and the soil movements measured at the Arlington site over two years (modified after Zhang 2004)........

71

2.27 Soil water content versus volumetric strain obtained from the shrink test (modified after Briaud et al. 2003)........................................................................

73

2.28 Soil movements predicted by Briaud et al. (2003) method and the measured soil movements at the Arlington site over two years (modified after Briaud et al. 2003).................................................................................................................

74

2.29 Predicted and measured monthly surface movements at 1.8 m outside slab edge along the longitudinal axis at Amarillo site (modified after Wray et al. 2005).....

78

xii

3.1 The relationship between (a) soil-water characteristic curve (SWCC), (b) the variation of modulus of elasticity with respect to matric suction (modified after Oh et al. 2009).......................................................................................................

85

3.2 Relationship between 1/α and plasticity index Ip (modified after Vanapalli and Oh 2010)................................................................................................................

87

3.3 Comparison of typical stress-strain curve with hyperbolic stress-strain curve (modified after Al-Shayea et al. 2001)..................................................................

89

3.4 Transformed hyperbolic stress-strain curve (modified after Duncan and Chang

1970)......................................................................................................................

89

3.5 Soil-water characteristic curves (SWCCs) for the three investigated expansive soils (Zao-Yang, Nanyang, and Guangxi expansive soils)....................................

90

3.6 Stress-strain curves for specimens of saturated compacted Zao-Yang soils at various confining stresses (modified after Zhan 2003).........................................

92

3.7 Stress-strain curves for specimens of unsaturated compacted Zao-Yang soils: (a) at net confining stress of 50 kPa, (b) at net confining stress of 200 kPa (modified after Zhan 2003)...................................................................................

93

3.8 Stress-strain curves for specimens of saturated compacted Nanyang soils (modified after Miao et al. 2002)..........................................................................

94

3.9 Stress-strain curves for specimens of unsaturated compacted Nanyang soils: (a) at matric suction of 50 kPa, (b) at matric suction of 80 kPa, (c) at matric suction of 120 kPa, and (d) at matric suction of 200 kPa (modified after Miao et al. 2002).............................................................................................................

96

3.10 Stress-strain curves for specimens of saturated compacted Guangxi soils (modified after Miao et al. 2007)..........................................................................

97

3.11 Stress-strain curves for specimens of unsaturated compacted Guangxi soils: (a) at degree of saturation of 76.3%, (b) at degree of saturation of 83.5%, and (c) at degree of saturation of 92.1% (modified after Miao et al. 2007)..........................

98

3.12 Transformed stress-strain curve for specimens of saturated, compacted Zao-Yang soils..............................................................................................................

99

3.13 Transformed stress-strain curves for specimens of unsaturated, compacted Zao-Yang soils: (a) at net confining stress of 50 kPa, (b) at net confining stress of 200 kPa..............................................................................................................

100

3.14 The relationship of the saturated modulus of elasticity with the confining stress for specimens of compacted Zao-Yang soils.........................................................

101

xiii

3.15 Comparison between the experimental and predicted modulus of elasticity for Zao-Yang expansive soils......................................................................................

105

3.16 Comparison between the experimental and predicted modulus of elasticity for Nanyang expansive soils.......................................................................................

106

3.17 Comparison between the experimental and predicted modulus of elasticity for Guangxi expansive soils........................................................................................

106

3.18 Predicted moduli versus experimental moduli for the three investigated expansive soils (Zao-Yang, Nanyang, and Guangxi expansive soils)...................

107

3.19 The plot of (1/α) versus the plasticity index Ip for the three investigated expansive soils along with the upper and lower boundary relationships of (1/α) versus Ip proposed by Vanapalli and Oh (2010)...................................................

108

3.20 The plot of (1/α) versus net confining stress 3( )auσ − for the three investigated expansive soils (Zao-Yang, Nanyang, and Guangxi expansive soils)...................

109

4.1 Capillary pressure and swelling process (modified after Terzaghi 1931)............. 113

4.2 Flowchart for the step-by-step procedure of the MEBM...................................... 122

4.3 Three step-procedure of the MEBM...................................................................... 124

5.1 Geometry and boundary conditions of Case Study A (modified after Vu and Fredlund 2006)......................................................................................................

128

5.2 Hydraulic characteristics of Regina expansive clay used for Case Study A (SWCC data obtained from Vu 2002)...................................................................

128

5.3 Matric suction changes with time for the three locations A, B, and C.................. 130

5.4 Matric suction profiles for various elapsed times at the right of the outer edge of the slab..............................................................................................................

130

5.5 Oedometer test results for Regina expansive clay along with the best fit equations ...............................................................................................................

133

5.6 Comparison between the predicted heaves using the proposed MEBM and Vu and Fredlund (2006) method at three locations A, B, and C.................................

136

5.7 Geometry and boundary conditions of Case Study B (modified after Vu and Fredlund 2004)......................................................................................................

138

5.8 Hydraulic characteristics of unsaturated Regina expansive clay for Case Study B (modified after Vu and Fredlund 2004).............................................................

139

5.9 Matric suction changes with time for the three locations D, E, and F.................. 140

xiv

5.10 Matric suction profiles for various elapsed times under the center of the slab..... 141

5.11 Predicted and measured soil heave profiles under the center of the slab.............. 142

5.12 Predicted and measured soil heave values along the surface of the slab.............. 143

5.13 Schematic of the test site of Case Study C (modified after Ito and Hu 2011)...... 146

5.14 Soil profile and soil properties of Case Study C (modified after Ito and Hu 2011)......................................................................................................................

146

5.15 Soil-water characteristic curves of the site soils for Case Study C (modified after Ito and Hu 2011)...........................................................................................

147

5.16 Permeability functions of the site soils for Case Study C (modified after Ito and Hu 2011)................................................................................................................

147

5.17 Climate data for the Regina test site of Case Study C (modified after Ito and Hu 2011)................................................................................................................

149

5.18 Soil matric suction changes with respect to time at different depths under the centre point of the vegetation cover for Case Study C..........................................

151

5.19 Volumetric water content changes with respect to time at different depths under the centre point of the vegetation cover for Case Study C..........................

152

5.20 Predicted matric suction profiles using VADOSE/W at different times under the centre point of the vegetation cover for Case Study C....................................

152

5.21 Predicted vertical movements of clay layer for each day at different depths below the centre of the vegetation cover using the MEBM..................................

154

5.22 Predicted total accumulated vertical soil movements at different depths under the centre of the vegetation cover using the MEBM.............................................

155

5.23 Comparison of the vertical soil movement for each day at the ground surface and the 0.5 m depth predicted using the MEBM and Ito and Hu (2011) model...

157

5.24 Cross-section of the instrumented slope of Case Study D (modified after Ng et al. 2003).................................................................................................................

159

5.25 Cracks and fissures in an excavation pit near the monitoring area of Case Study D (Zhan et al. 2007)...............................................................................................

160

5.26 Soil-water characteristic curves for the slope soil (modified after Zhan et al. 2006)......................................................................................................................

160

5.27 Intensity of rainfall events during the monitoring period of Case Study D (modified after Ng et al. 2003)..............................................................................

161

xv

5.28 2-D model for the research slope of Case Study D............................................... 162

5.29 Key soil properties of the 4-layers considered for the research slope: (a) 4-soil layers considered, (b) soil-water characteristic curves, and (c) coefficient of permeability functions of the slope soil.................................................................

165

5.30 Comparison of the predicted and the measured pore water pressure (PWP) during the rainfall events at different depths at the mid-slope..............................

165

5.31 Comparison of the predicted and the measured VWC during the rainfall events at different depths at the mid-slope.......................................................................

168

5.32 Matric suction profiles at the middle of the slope for Case Study D.................... 169

5.33 Comparison of the estimated vertical soil movements with the field measurements at the mid-slope.............................................................................

170

5.34 Comparison of the estimated soil heaves using the MEBM with the field measurements at the mid-slope.............................................................................

172

5.35 Soil stratigraphy and basic soil properties for the site of Case Study E (modified after Briaud et al. 2003)........................................................................

173

5.36 Soil-water characteristic curves: (a) for dark gray silty clay, (b) for brown silty clay (modified after Briaud et al. 2003)................................................................

174

5.37 Permeability functions estimated by VADOSE/W for the soil layers of Case Study E..................................................................................................................

174

5.38 Daily climate data over two years for the Arlington site of Case Study E (modified after Zhang 2004).................................................................................

175

5.39 Measured movements of the footings at the Arlington site over two years (modified after Briaud et al. 2003)........................................................................

176

5.40 The soil domain for Case Study E along with the initial and the boundary conditions used for the simulation of matric suction changes over time (modified after Zhang 2004).................................................................................

177

5.41 Typical leaf area index function for excellent grass coverage (modified after GeoSlope 2007).....................................................................................................

178

5.42 Comparison between the initial water content profiles obtained from VADOSE/W and Zhang (2004)............................................................................

179

5.43 The variations of the average value of the predicted and the measured water contents for the four footings over the 3 m depth with respect to time.................

179

xvi

5.44 Matric suction profiles at the corner of the footing simulated using VADOSE/W..........................................................................................................

180

5.45 Oedometer test results and their fitting curves for the specimens of dark gray silty clay and brown silty clay at the Arlington site (modified after Zhang 2004)......................................................................................................................

181

5.46 Variation of the saturated modulus of elasticity with depth for the investigated soils at the Arlington site.......................................................................................

181

5.47 Comparison between the predicted soil movements at the corner of the modeled footing using the MEBM and the field measurements of the soil movement for the four footings.............................................................................

183

5.48 Comparison of the soil movement predicted using different methods with the average field measurements of soil movement for the four footings....................

184

6.1 The relationship of the dimensionless parameter X versus ( )unsat sat aE E P− for compacted specimens of Zao-Yang soils tested under different confining stresses...................................................................................................................

194

6.2 Comparison between the experimental and the estimated values of the unsaturated modulus of elasticity for Zao-Yang soil............................................

195

6.3 The relationship of X versus ( )unsat sat aE E P− for compacted specimens of Nanyang soil tested under different confining stresses.........................................

197

6.4 Comparison between the experimental and the estimated values of the unsaturated modulus of elasticity for Nanyang soil..............................................

198

6.5 The relationship of X versus ( )unsat sat aE E P− for compacted specimens of Guangxi soil tested under different confining stresses..........................................

199

6.6 Comparison between the experimental and estimated values of the unsaturated modulus of elasticity for Guangxi soil..................................................................

200

6.7 Comparison between the values of elasticity moduli estimated from the proposed dimensionless model and the VO model for the three investigated expansive soils.......................................................................................................

201

6.8 Comparison between the values of elasticity moduli obtained from the dimensionless model and the triaxial tests for the three investigated expansive soils........................................................................................................................

202

6.9 Comparison of the predicted soil movements using the MEBM based on the dimensionless model with the field measurements at the mid-slope....................

204

xvii

LIST OF TABLES

1.1 The annual costs associated with the damage to structures constructed on expansive soils for different regions in the world.................................................

3

2.1 Factors influencing the magnitude and the rate of soil volume change (modified after Holtz and Gibbs 1956, Seed et al. 1962, Jennings 1969, Chen 1975, Johnson and Snethen 1978, Holland and Cameron 1981, Jones and Jefferson 2012)......................................................................................................

18

2.2 Examples of software packages commonly used for unsaturated flow modeling with atmospheric interactions (Benson 2007).......................................................

29

2.3 Effective stress equations for unsaturated soils (modified after Lu 2010)........ 30

2.4 The most common empirical methods for the determination of soil heave potential (modified after Rao et al. 2011 and Vanapalli and Lu 2012).................

56

2.5 Suction-based methods for predicting heave potential (modified after Vanapalli and Lu 2012)..........................................................................................................

60

2.6 Summary of the current methods for predicting the volume change movement of expansive soils over time..................................................................................

81

3.1 Soil properties of Zao-Yang, Nanyang, and Guangxi expansive soils.................. 91

3.2 Experimental elasticity moduli obtained from the triaxial tests for compacted Zao-Yang soils under the saturated and unsaturated conditions (data from Zhan 2003)......................................................................................................................

102

3.3 Experimental elasticity moduli obtained from the triaxial tests for compacted Nanyang soils under the saturated and unsaturated conditions (data from Miao et al. 2002).............................................................................................................

102

3.4 Experimental elasticity moduli obtained from the triaxial tests for compacted Guangxi soils under the saturated and unsaturated conditions (data from Miao et al. 2007)……………….....................................................................................

102

3.5 The fitting parameters and the predicted elasticity moduli estimated using the VO model for unsaturated, compacted Zao-Yang soils........................................

104

3.6 The fitting parameters and the predicted elasticity moduli estimated using the VO model for unsaturated, compacted Nanyang soils..........................................

104

xviii

3.7 The fitting parameters and the predicted elasticity moduli estimated using the VO model for unsaturated, compacted Guangxi soils..........................................

104

5.1 Case studies simulated using the proposed MEBM.............................................. 126

5.2 Mechanical properties of Regina expansive clay for Case Study A (modified after Shuai 1996)...................................................................................................

129

5.3 Fitting parameters of the void ratio constitutive surface for Regina expansive clay (Vu and Fredlund 2006)………………………….........................................

132

5.4 Soil properties for Case Study A (modified after Vu and Fredlund 2006)........... 134

5.5 Soil properties for Case Study B (Vu and Fredlund 2004).................................... 139

5.6 Soil properties for Case Study C (Ito and Hu 2011)............................................. 145

5.7 Physical properties of soil specimens taken from the research slope at a depth of 1.0 m (data from Zhan et al. 2007)....................................................................

159

5.8 Fitting parameters of the relationship of void ratio versus the mean stress for the investigated soils at the Arlington site (Zhang 2004)......................................

181

6.1 Experimental data and the corresponding dimensionless parameter X for the compacted specimens of Zao-Yang soils under unsaturated conditions (data from Zhan (2003)).................................................................................................

194

6.2 Experimental data and the corresponding dimensionless parameter X for the compacted specimens of Nanyang soil under unsaturated conditions (data from Miao et al. (2002)).................................................................................................

196

6.3 Experimental data and the corresponding dimensionless parameter X for the compacted specimens of Guangxi soil under unsaturated conditions (data from Miao et al., 2007)...................................................................................................

199

xix

SUBSCRIPTS

f = final value

i = initial value, or order

max = maximum value

mean = mean value

min = minimum value

unsat = unsaturated condition

sat = saturated condition

ABBREVIATIONS AND SYMBOLS

a, b, c, d, f, g = fitting parameters of the void ratio constitutive surface proposed by Vu and

Fredlund (2006)

ma = coefficient of compressibility with respect to change in matric suction

ta = coefficient of compressibility with respect to change in net normal stress

wa = ratio of area of water-mineral and water-water contact of total area of “wavy”

plane

1a , 1b , 1x , 1y = fitting parameters of the relationship of void ratio versus net normal stress for

saturated soils proposed by Zhang (2004)

cA = soil activity

AEV = air-entry value

mb = coefficient of water content change with respect to a change in matric suction

jb = components of body force vector

tb = coefficient of water content change with respect to a change in net normal stress

LIST OF SYMBOLS

xx

B = slope of suction versus water content relationship

C = clay content

cC = compression index

hC = suction index with respect to void ratio

HC = heave index in Nelson and Miller (1992) method

mC = compressive index with respect to matric suction

Cs = swelling index

tC = compressive index with respect to total stress

wC = specific water capacity of a soil, or suction modulus ratio used by Vanapalli et

al. (2010)

Cτ , Cψ = suction index

COLE = coefficient of linear extensibility

D = independent primary dimension in the dimensional analysis

e = void ratio

0e = initial void ratio

ie = void ratio for the ith layer

fe = final void ratio

E = initial tangent modulus of elasticity

Eunsat = soil modulus of elasticity under unsaturated condition

Esat = soil modulus of elasticity under the saturated condition

wE = water volumetric modulus associated with a change in net normal stress, or

shrink-swell modulus proposed by Briaud et al. (2003)

f = crack fabric factor proposed by Lytton et al. (2004), function, shrinkage ratio, or

lateral restraint factor proposed by McKeen (1992)

( , , , )f x y z t = internal source of moisture in the transient suction diffusion equation developed

by Mitchell (1979)

if = lateral confinement factor proposed Lytton (1977)

xxi

iF = initial state factor in soil heave potential equation proposed by Zumrawi (2013)

FSI = free swell index

g = gravitational acceleration

G = shear modulus

sG = specific gravity

h = soil water pressure head

0h = initial height of a specimen tested in triaxial shear tests

ih , fh = initial and final water potentials, or initial and final matric suction

ih , iH , iz∆ , t∆ = thickness of the ith soil layer

mh = matric suction

sh = solute suction

H = total hydraulic head, modulus of elasticity for the soil structure with respect to a

change in matric suction, depth of soil, or soil layer thickness

wH = water volumetric modulus associated with a change in matric suction

i = hydraulic gradient

ptI = instability index

IL = liquidity index

k = coefficient of permeability (i.e., hydraulic conductivity)

wik = coefficient of permeability in the i direction

ksat = saturated coefficient of permeability

K = fitting parameter in Janbu’s relationship (1963) for the soil modulus of

elasticity, or correction parameter used by Vanapalli et al. (2010)

0K = coefficient of lateral earth pressure at rest

l = number of variables in the dimensional analysis

L = dimension of length in the dimensional analysis

LL , wL = liquid limit

WLL = weighted liquid limit

m = number of independent primary dimensions in the dimensional analysis

xxii

1sm = coefficient of total volume change with respect to change in net normal stress

2sm = coefficient of volume change with respect to change in matric suction

1wm = coefficient of water volume change with respect to change in net normal stress

2wm = coefficient of water volume change with respect to change in matric suction

M = dimension of mass in the dimensional analysis

MBV = methylene blue value proposed by Cokca (2002)

n = total number of soil layers considered, or fitting parameter in Janbu’s

relationship (1963) for the soil modulus of elasticity

N = number of dimensionless parameters in the dimensional analysis

p = unsaturated permeability in the transient suction diffusion equation developed

by Mitchell (1979)

p′′ = pore water pressure deficiency proposed by Donald (1956)

mp′′ = matric suction in the effective stress equations for unsaturated soils proposed by

Aitchison (1973)

sp′′ = solute suction in the effective stress equation for unsaturated soils proposed by

Aitchison (1973)

P = partial pressure of pore water vapor, or surcharge pressure

0P = saturation pressure of water vapour over a flat surface of pure water

Pa , atmu = atmospheric pressure

fP = final stress state

Ps = corrected swelling pressure

PI , Ip = plasticity index

PL, wp = plastic limit

q = surcharge pressure

iq = initial surcharge pressure

R = osmotic suction in the effective stress equation for unsaturated soils proposed

by Allam and Sridharan (1987), or universal gas constant (8.31432 J/(mol K))

hR = relative humidity

xxiii

sR = radius of curvature of the water meniscus

2R = coefficient of determination

s = reduction factor to account for overburden proposed by McKeen (1992)

S = degree of saturation, or heave/swell potential

pS = heave/swell potential

fS = surface displacement

SI = shrinkage index

SP = swell pressure applied to the soil due to overburden pressure

SW = percent swell

u = total suction

au = pore-air pressure

cu = capillary pressure

iu = components of displacement in the i-direction

wu = pore-water pressure

t = time

T = temperature, or dimension of time in the dimensional analysis

sT = surface tension

wv = Darcy’s flux (flow rate per unit area)

V = a variable in the dimensional analysis

0V = initial overall volume of an unsaturated soil element

aV = volume of air in an unsaturated soil element

molV = molecular volume of water vapour (0.01802 m3)

sV = volume of solid particles in an unsaturated soil element

vV = volume of soil voids in an unsaturated soil element

voV = volume of voids in an unsaturated soil element

wV = volume of water in an unsaturated soil element

iw , 0w = initial water/moisture content

xxiv

fw = final water/moisture content

nw = natural moisture content

woptm = optimum water content

x , y , z = space coordinates

X = dimensionless parameter in the dimensional analysis

Y = elevation

z = soil depth

α = compressibility index proposed by Johnson and Snethen (1978), diffusion

coefficient, or fitting parameter in Vanapalli and Oh (2010) model

β = fitting parameter in Vanapalli and Oh (2010) model

1β = statistical factor of the same type as the contact area in the effective stress

equation for unsaturated soils proposed by Jennings (1961)

β ′ = holding or bonding factor in the effective stress equation for unsaturated soils

proposed by Croney et al. (1958)

dγ = dry unit weight of soil

diγ = initial dry unit weight of soil

γdmax = maximum dry unit weight of soil

hγ = suction compressibility index proposed by Wray (1984)

tγ = total unit weight of soil

wγ = unit weight of water

yzγ , zxγ , γ xy = shear strains on the x-, y-, and z-planes

σγ = mean principal stress compression index

δ = vertical shrinkage or heave

ijδ = Kroneker delta or substitution tensor in the effective stress equation for

unsaturated soils proposed by Jommi (2000)

∆c = difference in the concentration between two solutions

h∆ = total vertical movement (heave/shrink) at any depth, or change in the specimen

height upon compression

xxv

ih∆ = vertical movement for each arbitrary layer

H∆ = soil heave, heave/swell potential, or surface displacement

pF∆ = change in the soil suction over a vertical increment

pP∆ = change in the soil overburden over a vertical increment

u∆ = soil suction change

w∆ = moisture content change

yε∆ = change in vertical strain

ε = axial strain, or mean-zero Gaussian random error term proposed by Türköz and

Tosun (2011)

ijε = components of the strain tensor

vε = volumetric strain

xε , yε , zε = normal strains in the x-, y-, and z-directions respectively

ς = interface parameter on the reference plane in the effective stress equation for

unsaturated soils proposed by Allam and Sridharan (1987)

θs = saturated volumetric water content

wθ = volumetric water content

λ = parameter proposed by Nelson et al. (2006)

µ = Poisson’s ratio

π = osmotic suction, or dimensionless parameter (i.e. Pi group) in the dimensional

analysis

dρ = dry density of soil

iρ = maximum heave

wρ = density of water

σ = externally applied stress

iσ , fσ = initial and final values of mean principal stress

ijσ = total stress tensor

*ijσ = Bishop’s average soil skeleton stress

xxvi

meanσ = mean total normal stress

xσ , yσ , zσ = normal stresses in the x-, y-, and z-directions, respectively

1σ , 3σ = major and minor principal stresses, respectively

′σ = effective stress

csσ ′ = swelling pressure from the overburden swell test

cvσ ′ = swelling pressure from constant volume oedometer test

iσ ′ = inundation stress subjected to a sample in overburden swell test

'obσ = overburden stress

ϕ = parameter in the effective stress equation for unsaturated soils proposed by

Aitchison (1960)

χ = effective stress parameter related to the degree of soil saturation

mχ = effective stress parameter for matric suction

sχ = effective stress parameter for solute suction

ψ = total suction

iψ , fψ = initial and final total suction, respectively

xxvii

CHAPTER 1

INTRODUCTION

1.1 Background

Expansive soils formation may probably be associated to the gradual weathering or

erosion of basic igneous rocks or sedimentary rocks (Donaldson 1969). The minerals of

the parent rocks from which expansive soils are derived decompose to form the highly

active clay minerals (e.g., montmorillonite). These minerals have an affinity for

absorbing large amounts of water between their clay sheets and therefore have a large

shrink–swell potential. When potentially expansive soils become saturated, after rainfall

or due to other activities (e.g., garden watering, or water pipe leaks), more water

molecules are absorbed between the clay sheets, causing the bulk volume of the soil to

increase or swell (heave). Conversely, when water is removed, by evaporation or

gravitational forces, the water between the clay sheets is released, causing the overall

volume of the soil to decrease or shrink (Jones and Jefferson 2012). When supporting

structures, the effect of significant changes in the water content of expansive soils can be

severe. The heave/shrink movements of expansive soils can cause tilt in trees, highway

surfaces, building foundations and pipelines, and pose problems to the functionality of

the infrastructure. These soils have also been referred to in the literature with various

names such as the swelling soils, heaving soils, volume change soils, shrink-swell

soils, problematic soils, or black cotton soils.

Expansive soils are vastly distributed in many regions around the world, and their

distribution is dependent on many factors such as the geology, climate, hydrology,

geomorphology, and vegetation of the region. The countries in which expansive soils

have been reported are: Argentina, Australia, Burma, Canada, China, Cuba, Ethiopia,

1

Ghana, Great Britain, India, Iran, Kenya, Mexico, Morocco, Rhodesia, South Africa,

Spain, Turkey, USA, and Venezuela (Chen 1975, Fredlund and Rahardjo 1993).

Approximately 60% of the world’s population live in regions with expansive soils and

have no choice but to construct their infrastructures on these problematic soils.

Problems associated with expansive soils were first recognized and documented by the

U.S. Bureau of Reclamation at their Owyhee Project in Oregon in 1930 (Holtz and Gibbs

1954). Research interest in expansive soils to better understand them has increased as

more extensive damages to structures were being documented after 1940. Jones and

Holtz (1973) were the first investigators who estimated the damages associated with

expansive soils. The damages to lightly loaded structures constructed in the United States

alone amounts to about $2.3 billion per year. Krohn and Slosson (1980) estimates show

the losses to the structures increased to about $7 billion per year. Steinberg (1998)

reported further increases to these losses and reported them to be approximately $10

billion annually. More recently, Puppala and Cerato (2009) reported that the cost of

damage due to expansive soils in the United States has risen dramatically to over $13

billion per year. Figure (1.1) shows that the losses associated with expansive soils in the

United States alone have increased five fold during the last four decades. It is worth that

the cumulative deleterious effects that expansive soils have on constructed facilities in the

United States annually exceeds those of hurricanes, tornadoes, floods, and earthquakes

combined (Jones and Holtz 1973). Expansive soils have been called the “hidden

disaster”: while they do not typically cause loss of life, economically, they are one of the

United States costliest natural hazards (Snethen 1986). Similar problems have been

reported in many other countries. Table (1.1) shows the annual costs associated with the

infrastructure damage problems caused by the movements of expansive soils in different

regions of the world. It is no wonder that these soils are considered to be a nightmare by

geotechnical engineers.

CHAPTER 1 2

Fig. 1.1. The annual costs of damage to structures constructed on expansive soils in the United States since 1973

Table 1.1. The annual costs associated with the damage to structures constructed on

expansive soils for different regions in the world

Region Cost of damage/ year Reference USA $13 billion Puppala and Cerato (2009) UK £ 400 million Driscoll and Crilly (2000) France € 3.3 billion Johnson (1973) Saudi Arabia $ 300 million Ruwaih (1987) China ¥ 100 million Ng et al. (2003) Victoria, Australia $ 150 million Osman et al. (2005)

Expansive soils are prone to heave (swelling) and shrinkage as they are sensitive to even

small changes in their natural water content conditions. The volume change behavior of

expansive soils is one of the key engineering properties that influence the performance of

lightly loaded structures placed on expansive soils. Therefore, it is important to provide

tools for the practitioners to reliably estimate the heave and shrink related volume change

of expansive soils. Several methods have been proposed in the literature for the

determination, or prediction of the volume change movement of expansive soils

(Vanapalli and Lu 2012). The methods developed can be classified into three main

categories namely; empirical methods, oedometer methods, and suction based methods.

The focus has been towards estimating the maximum potential heave (i.e., the extreme

INTRODUCTION 3

condition), which occurs when soils attain the saturation condition. However, the results

of heave estimation considering saturated soil conditions are not always practical or

economical in engineering practice. A reliable design of lightly loaded structures on

expansive soils is likely if the anticipated soil movements in the field can be estimated

over time, taking into account the influence of environmental changes. Limited studies

were reported in the literature to measure the soil movements in the field during the past

three decades (Yoshida et al. 1983, Ching and Fredlund 1984, Clifton et al. 1984, Wray

1989, Allman et al. 1998, Erol and Dhowian 1990, Ng et al. 2003, Briaud et al. 2003,

Fityus et al. 2004, Chao 2007, Tang et al. 2009). More studies in this direction are useful

to better understand the behavior of expansive soils with respect to time. However, they

are expensive and cumbersome and hence cannot be undertaken for routine engineering

practice.

During the past decade focus of research has been towards proposing prediction

procedures for estimating the expansive soil movement over time (Briaud et al. 2003, Vu

and Fredlund 2004 and 2006, Zhang 2004, Wray et al. 2005, Overton et al. 2006, Nelson

et al. 2007). The proposed methods, however, suffer from the need to run expensive and

time consuming tests, and limited verification studies for different expansive soils. In

other words, the methods validity was limited to expansive soils of local regions and has

not been extended widely. The prediction of volume change movements is a challenge

even to date to geotechnical engineers as heave and shrinkage behavior with respect to

environmental changes cannot be well estimated.

1.2 Objective

The key objective of this research is to develop a simple and efficient method, which is

referred to as a modulus of elasticity based method (MEBM), for predicting the vertical

movements of natural expansive soils associated with the variations of environmental

conditions. The research program focus is directed towards: (i) developing a simple

constitutive relationship for estimating the vertical movements with respect to time in

terms of the matric suction variations and the corresponding modulus of elasticity; (ii)

modelling the soil-atmospheric interactions considering the environmental variations to

CHAPTER 1 4

simulate the matric suction changes over time; (iii) developing simple models to estimate

the soil modulus of elasticity with respect to matric suction.

While all the existing prediction methods published to date were limited to localized

areas, the proposed MEBM is tested for its validity in several case studies collected and

gathered from the literature. These case studies, chosen to be representative of a variety

of site conditions from different regions of the world, include:

- A slab-on-ground placed on Regina expansive clay subjected to a constant

infiltration rate, which was originally modeled by Vu and Fredlund (2006).

- A case history of a light industrial building in North-Central Regina, Saskatchewan,

Canada. History of the site and details of testing and monitoring programs were

conducted by Yoshida et al. (1983). Heave analyses of the case history using

laboratory oedometer data were carried out by Vu and Fredlund (2004).

- A field test site in Regina, Saskatchewan, modeled by Ito and Hu (2011) for one

year. Various factors influencing soil movements such as climate changes,

vegetation, watering of lawn, and soil cover type have been considered.

- A comprehensive field study previously investigated by Ng et al. (2003). This field

study is a cut-slope in an expansive soil in Zao-Yang, Hubie, China, in which the

effect of the soil compressibility, soil cracks, and environmental conditions on the

soil movements have been investigated.

- A field experiment in Arlington, Texas, conducted by Briaud et al. (2003) for

measuring the movements of four full-scale spread footings over a period of 2 years.

The MEBM approach proposed in this thesis is assessed by providing comparisons

between soil movement estimates and the published results of the case studies under

consideration.

1.3 Research Methodology

The research methodology can be summarized by the following:

INTRODUCTION 5

- Review the available literature on the expansive soils to identify the most significant

soil properties directly related to the volume change behavior of expansive soils.

- Develop basic theoretical understanding of the state-of-the-art prediction methods of

the volume change of expansive soils, and critically review the current prediction

methods in terms of their predictive capacities and their strengths and limitations for

their use.

- Extend Vanapalli and Oh (2010) model, originally developed for estimating the

modulus of elasticity in terms of matric suction for fine-grained soils with plasticity

index Ip values lower than 16%, to be used for expansive soils (i.e., Ip > 16%) and to

test its validity using experimental data of triaxial shear tests for different expansive

soils from the literature.

- Develop a constitutive relationship based on the volume change theory of

unsaturated soils to predict the vertical movements of natural expansive soils over

time.

- Simulate the time-evolution of matric suction profile within the active zone of soil

based on the numerical modeling of the soil-atmospheric interactions. A finite

element program VADOSE/W (Geo-Slope 2007) is used in this research study for

this purpose.

- Apply the step-by-step procedure of the proposed modulus of elasticity based

method (MEBM) on different case studies summarized in the preceding section to

test its validity for the prediction of the heave and shrink movements of unsaturated

expansive soils with respect to time.

- Propose a dimensionless model for estimating the modulus of elasticity of

unsaturated expansive soils based on the dimensional analysis of experimental data

of triaxial tests for different expansive soils, taking into account all the influencing

parameters.

- Revisit a field study conducted by Ng et al. (2003) to evaluate the proposed MEBM

based on using the new dimensionless model for estimating the modulus of elasticity

of unsaturated expansive soils.

CHAPTER 1 6

1.4 Novelty of the Research Study

The mechanics of unsaturated soils has been used as a tool to interpret the heave and

shrink related volume change behavior of expansive soils since 1970’s. However, there

are few methods to predict the volume change behavior of unsaturated expansive soils

over time. Existing prediction methods suggest that a given soil will exhibit a unique

three-dimensional surface relating void ratio (i.e., volume change) to the mechanical

stress and matric suction (Vu and Fredlund 2004, Zhang 2004). However, there are

limitations to apply the three-dimensional constitutive surface model in practice. Current

volume change constitutive surfaces are developed based on testing soils under

conditions not experienced in the field such as a shrinkage test or a matric suction test at

no normal stress, or a consolidation test at fully saturated conditions. Also, many

conventional laboratories are not equipped to run controlled matric suction tests. These

tests generally require costly, time consuming, and difficult laboratory testing. Most

importantly, the prediction methods based on the volume change constitutive surface

have been only validated for one case study.

The innovative aspect of the research presented in this thesis is to predict the volume

change movement of natural expansive soils for different case studies using one simple

approach (i.e., modulus of elasticity based method (MEBM)). The proposed MEBM is

based on soil properties determined by using conventional geotechnical testing methods.

This is the first time in the literature that a simplified constitutive relationship for the total

volume change of soil is used to estimate the vertical soil movements in terms of the soil

suction variations and the associated modulus of elasticity.

The pioneering work by Terzaghi (1925, 1926, and 1931) to understand the shrinkage

and swelling behavior of clay showed that the shrinkage and swelling capacity of any soil

are essentially dependent on the elastic properties of the solid phase of the soil. These

fundamental studies of Terzaghi though not as widely cited as his other research studies

in the conventional geotechnical literature, have significantly contributed to our present

state-of-the-art interpretation of the volume change movements of expansive soils in

terms of the soil modulus of elasticity. Vanapalli and Oh (2010) proposed a semi-

INTRODUCTION 7

empirical model for estimating the variation of the modulus of elasticity with respect to

matric suction for soils with plasticity index Ip values lower than 16%. This model has

been extended in this study to estimate the modulus of elasticity of unsaturated expansive

soils (i.e., Ip > 16%). The information required for using the model include the soil-water

characteristic curve (SWCC) and the modulus of elasticity of soil under saturated

condition along with two fitting parameters. Experimental data of triaxial tests for three

different expansive soils from the literature are used in this study to examine the validity

of the model for expansive soils. Good comparisons are provided between the values of

modulus of elasticity derived from triaxial tests results and from the modified Vanapalli

and Oh (2010) model.

Based on the dimensional analysis of the same experimental data of triaxial tests used for

the Vanapalli and Oh (2010) model, a new innovative model is proposed in this study to

estimate the modulus of elasticity of unsaturated expansive soils. The new dimensionless

model provides a more realistic characterization of the soil modulus of elasticity, taking

account of the influence of matric suction and mechanical stress along with initial void

ratio and degree of saturation.

The proposed MEBM is tested for its validity in five case studies from three countries:

Canada, China, and the United States for a wide variety of site and environmental

conditions. The MEBM, in comparison to other available methods, is simple and efficient

for the prediction of vertical movements of natural expansive soils over time. The

strength of the MEBM lies in its use of available soil properties that can be determined by

using conventional geotechnical testing methods. The results of the research study are

valuable to provide guidelines for design of structures constructed on expansive soils.

1.5 Layout of the Thesis

The thesis is organised into seven chapters. This chapter presents the problem definition,

objective, methodology and novelty of the research study, and the layout of the thesis.

CHAPTER 1 8

Chapter Two provides literature review that includes a comprehensive and detailed

description on background of the expansive soils that is necessary for explaining the

research studies presented in the thesis. The focus of the chapter has been directed to

summarize the various approaches available in the literature for the prediction of volume

change behavior of expansive soils.

Chapter Three provides an evidence for the validity of Vanapalli and Oh (2010) model to

predict the modulus of elasticity for expansive soils under variably saturated conditions.

Available experimental data of triaxial shear tests for different compacted unsaturated

expansive soils is used in the chapter to examine the validity of the Vanapalli and Oh

(2010) model.

Chapter Four details the fundamental concepts along with the step-by-step procedure of

the proposed modulus of elasticity based method (MEBM) for predicting the vertical

movement of natural expansive soils over time. Variations of soil matric suction and the

corresponding modulus of elasticity with respect to matric suction variations are

introduced into a volume change constitutive relationship to estimate the soil movement

with respect to time. The soil-atmosphere model VADOSE/W is selected to be used for

modeling the matric suction variations associated with the environmental changes over

time for all case studies simulated in this research.

Chapter Five presents the validation of the MEBM approach for predicting the vertical

soil movements over time using different case studies. Each case study is described, and

the soil properties used in the analysis are listed. A detailed description of the simulation

of each case study is also presented in this chapter. Comparisons of the results of the

MEBM with the published data (measurements/estimates) are provided.

Chapter Six proposes an alternative model for estimating the unsaturated modulus of

elasticity based on the dimensional analysis of the triaxial shear test results of compacted

unsaturated expansive soils. The model takes into account the most significant factors

influencing the value of the modulus of elasticity of unsaturated expansive soils.

Comparisons are provided between the values of modulus of elasticity derived from the

triaxial tests results, the extended Vanapalli and Oh (2010) model, and the new

INTRODUCTION 9

dimensionless model. In addition, a field study previously investigated by Ng et al.

(2003) is revisited in this chapter to evaluate the MEBM using the dimensionless model

as a tool for estimating the soil modulus of elasticity.

Finally, conclusions, and recommendations and suggestions for future research studies

are presented in Chapter Seven.

Research undertaken through the present study has resulted in the

following peer review journal publications:

- Adem, H.H. and Vanapalli, S.K. 2014. A state-of-the art review of methods for

predicting the in situ volume change movement of expansive soil over time

(tentatively accepted for publication in the Journal of Rock Mechanics and

Geotechnical Engineering, revised version submitted).

- Adem, H.H. and Vanapalli, S.K. Heave prediction in a natural unsaturated expansive

soil deposit under a lightly loaded structure (submitted to Geotechnical and

Geological Engineering Journal).

- Adem, H.H. and Vanapalli, S.K. 2014. Elasticity moduli of expansive soils from

dimensional analysis. Geotechnical Research, 1(2): 60-72, DOI:

10.1680/gr.14.00006

- Adem, H.H. and Vanapalli, S.K. 2014. Soil-environment interactions modeling for

expansive soils. Environmental Geotechnics, DOI: 10.1680/envgeo.13.00089

- Adem, H.H. and Vanapalli, S.K. 2014. Prediction of the modulus of elasticity of

compacted unsaturated expansive soils. International Journal of Geotechnical

Engineering, DOI: http://dx.doi.org/10.1179/1939787914Y.0000000050

- Adem, H.H. and Vanapalli, S.K. 2013. Constitutive modeling approach for

estimating the 1-D heave with respect to time for expansive soils. International

Journal of Geotechnical Engineering, 7(2): 199-204.

CHAPTER 1 10

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

Expansive soils absorb large quantities of water after rainfall or due to local site changes

(such as leakage from water supply pipes or drains), becoming sticky and heavy.

Conversely, they can also become stiff when dry, resulting in shrinking and cracking of

the ground. This hardening and softening is known as ‘shrink-swell’ behavior (Jones and

Jefferson 2012). When supporting lightly loaded structures, the effect of significant

changes in moisture content on soils with a high shrink-swell potential can be severe.

Hence, it is important to provide tools for practitioners to reliably estimate the volume

change behavior of expansive soils in the field. Significant advances were made during

the last half-a-century to understand the volume change behavior of expansive soils. This

chapter reviews the literature that discusses the properties of unsaturated expansive soils.

The chapter also presents the constitutive relations for the volume change of unsaturated

expansive soils as well as the flow laws for the soil moisture migration. In addition, a

state-of-the-art of the prediction methods of the volume change movement of expansive

soils is succinctly summarized. The available prediction methods are critically reviewed

in terms of their predictive capacities and their strengths and limitations. The review

highlights the need for prediction methods that are conceptually simple yet efficient for

use in conventional engineering practice for different types of expansive soils.

2.2 Expansive Soil Mineralogy

The expansion potential of any particular expansive soil is mainly determined by the

percentage and the type of clay minerals in the soil. The shape and structure of the clay

LITERATURE REVIEW 11

minerals are determined by their arrangement of their constituent atoms which form thin

clay crystals. The three most important clay minerals are montmorillonite, illite, and

kaolinite. Montmorillonite is the clay mineral that contributes to the expansive soil

problems.

The clay minerals are formed through a complicated process from a variety of parent

materials. These materials typically include feldspars, micas, and limestone. The

alteration process that takes place on land is referred to as weathering and that on the sea

floor or lake bottom as halmyrolysis. The alteration process includes disintegration,

oxidation, hydration, and leaching (Chen 1975). The formation of montmorillonite

requires extreme disintegration, strong hydration, and restricted leaching. When the

leaching is restricted, magnesium, calcium, sodium, and iron cations may accumulate in

the system. Thus, the formation of montmorillonite minerals is aided by an alkaline

environment, presence of magnesium ions, and a lack of leaching (Tourtelot 1973). Such

conditions are favorable in semi-arid regions, particularly where evaporation exceeds

precipitation. Under these conditions, enough water is available for the alteration process,

but the accumulated cations will not be removed by flush rain (Tourtelot 1973, Chen

1975). The parent minerals for the formation of montmorillonite often consist of

ferromagnesium minerals, calcic feldspars, volcanic glass, and many volcanic rocks

(Chen 1975). Bentonite is highly plastic, swelling clay composed primarily of

montmorillonite which has been formed by the chemical weathering of volcanic ash. The

properties of bentonite are familiar to most geotechnical engineers. Expansive clays are

commonly referred to as bentonitic soils.

There are two fundamental molecular structures as the basic unit of the lattice structure of

clay minerals. These are the silica tetrahedron and the alumina octahedron. The silica

tetrahedron is composed of a silicon atom surrounded tetrahedrally by four oxygen ions

as shown on Figure (2.1a). When each oxygen atom is shared by two tetrahedral, a plate-

shaped layer is formed (Figure 2.1b). However, the alumina octahedron is composed of

an aluminum atom surrounded octahedrally by six oxygen ions as shown in Figure (2.2a).

Similarly, when each aluminum atom is shared by two octahedrons, a sheet is formed

(Figure 2.2b) (Chen 1975, Mitchell 1976). The clay minerals are characterized by staking

CHAPTER 2 12

arrangement of sheets of these units and the manner in which two successive two- or

three-sheet layers are held together.

Fig. 2.1. A single silica tetrahedron and the sheet structure of silica tetrahedrons arranged in a hexagonal network (Mitchell and Soga 2005)

Fig. 2.2. A single octahedral unit and the sheet structure of the octahedral units (Mitchell and Soga 2005)

Kaolinite is a typical two layer mineral composed of alternating silica and octahedral

sheets to form what is called a 1 to 1 lattice structure as shown in Figure (2.3a). Bonding

between successive layers is by both van der Waals forces and hydrogen bonds. The

bonding is sufficiently strong that there is no interlayer swelling in the presence of water

(Mitchell and Soga 2005).

Montmorillonite is a three-layer mineral having a single octahedral sheet sandwiched

between two silica sheets to give a 2 to 1 lattice structure as shown in Figure (2.3b).

LITERATURE REVIEW 13

Bonding between successive layers is by van der Waals forces and by cations that

balance charge deficiencies in the structure. These bonds are weak and easily separated

by adsorption of water (Mitchell and Soga 2005).

Iillite has similar structure to that of montmorillonite, but some of silicon atoms are

replaced by aluminum; and, in addition, potassium ions are present between the silica

sheet and adjacent crystals (Figure 2.3c). Because of these differences, the illite structural

unit layers are relatively fixed in position, so that polar ions cannot enter readily between

them and cause expansion. Also the potassium ions between layers are not easily

exchangeable. In other words, the interlayer bonding by potassium is sufficiently strong

that the basal spacing of illite remains fixed in the presence of water (Mitchell and Soga

2005).

Fig. 2.3. Schematic diagrams of the structures of (a) kaolinite, (b) montmorillonite, (c) illite (Mitchell and Soga 2005)

Absorption of water by clays leads to expansion. Generally, the amount of expansion

depends on percentages and types of clay minerals in the soil (Chen 1975, Bell and

Culshaw 2001). For example, the presence of a relatively large amount of

montmorillonite tends to increase water intrusion within the mass of soil and then create

swelling, while illite and kaolinite are largely inert and resistant to water penetration.

Mitchell and Soga (2005) and Grim (1968) provide extensive details of mineralogy of

clay minerals and their influence on the engineering behavior of soils.

CHAPTER 2 14

2.3 Mechanism of Soil Swelling

The mechanism of soil swelling has been described by several researchers (e.g.,

Anderson et al. 1973, Chen 1975, Schafer and Singer 1976, Nelson and Miller 1992,

Stavridakis 2006). One proposed mechanism of soil swelling is associated with the

interlayer expansion of clay mineral montmorillonite. The molecular structure of

montmorillonite has a particular affinity to attract and hold water molecules between the

clay crystal sheets. When potentially expansive soils become saturated, the clay mineral

montmorillonite can absorb large amounts of water molecules into gaps between its clay

sheets. As more water is absorbed, the sheets are forced further apart, leading to an

increase in soil pressure or an expansion of soil volume (Figure 2.4). On the other hand,

intraparticle swelling due to orientation of water films around high charge density clays

may also contribute to swelling of soils. Clay particles are mainly flat and electrically

charged (usually negative). This high potential is concentrated on the surface of the clay

particles causing attraction of bipolar water molecules. Thus, an orientation of water

molecules on the clay surface is achieved. As water molecules are attracted to the clay

particles by negative charges, they push the clay particles apart, causing an expansion or

swelling of the soil (Figure 2.5).

Fig. 2.4. The intercalation of water molecules in the inter-plane space of montmorillonite (Taboada 2003)

LITERATURE REVIEW 15

Fig. 2.5. The orientation of water films around high charge density clays (Mitchell and Soga 2005)

2.4 Volume Change Movement of Expansive Soil

Expansive soils experience extensive volume changes (heave/shrink) when there is an

environmental change such as water content increase due to introduction of moisture,

pressure release due to excavation, and desiccation caused by temperature increase. If the

increase in soil volume is not restrained and soil has an opportunity to swell in all the

directions, the soil increases its volume in all the directions with the same amount (Figure

2.6a). However, for the soil in the field near the ground surface, the lateral expansion

might be to some extent prevented by the adjacent or the surrounding soil. The soil

expansion or swell predominantly occurs in the vertical direction (Figure 2.6b).

(a) (b)

Fig. 2.6. Type of expansive soil movements: (a) soil volumetric expansion in three directions when there is no restriction, (b) soil heave when the lateral movement is restricted

CHAPTER 2 16

Drying shrinkage of soils is caused by particles movements resulting from a severe loss

of pore water. Pore water could be removed from soil by a wide variety of mechanisms.

For example, excavation or other works that lower ground water levels, prolonged

periods of low rainfall, and low rainfall in combination with high water demand mature

trees can lead to a reduction in pore water, and then a reduction in soil volume (Chen

1975). The soil decreases its volume in the lateral and vertical directions. The volume

decrease in the vertical direction causes the soil surface to go down (shrink); the lateral

decrease in volume causes the soil to crack, if it is large enough.

On the other hand, based on the water infiltration and the type of structure, there are two

main types of expansive soil movements that could occur after construction and affect the

structure performance:

- Lateral movement

For many retaining and basement walls, especially if the clay backfill is

compacted below the optimum moisture content, seepage of water into the clay

backfill causes horizontal swelling pressures well in excess of at-rest values.

Lateral thrust of expansive soil with a horizontal force approaching the passive

earth pressure can cause bulging and fracture of basement and retaining walls.

While it is possible that a large amount of swelling pressure can be exerted

horizontally against a wall, generally backfill is so loosely compacted that distress

caused by lateral expansion of backfill is unlikely (Chen 1975).

- Vertical movement

If a structure having a large area, such as a pavement or foundation, is constructed

on top of expansive soil, there are usually two main types of soil movements. The

first is the cyclic heave and shrinkage around the perimeter of the structure,

related to the amount of drainage and the frequency and the amount of rainfall and

evaporation (edge uplift) (Figure 2.7a). The second is the long-term progressive

swell beneath the center of the structure, which can occur as an upward long term

dome shaped movement (center heave) (Figure 2.7b). Moisture can accumulate

underneath structures either by thermal osmosis or capillary action.

LITERATURE REVIEW 17

(a) Edge uplift

(b) Doming heave

Fig. 2.7. Vertical movement of expansive foundation soils: (a) edge uplift, (b) doming heave (modified after Department of the Army USA 1983)

2.5 Factors Affecting Soil Volume Change

The mechanism of soil volume change is complex and is influenced by a number of

factors which are not easy to quantify. The major factors can be classified into two broad

categories, as shown in Table (2.1): (i) in-situ soil properties and site conditions, (ii)

environmental influence arising from proposed land use (Holtz and Gibbs 1956, Seed et

al. 1962, Jennings 1969, Chen 1975, Johnson and Snethen 1978, Holland and Cameron

1981, Jones and Jefferson 2012).

Table 2.1. Factors influencing the magnitude and the rate of soil volume change

(modified after Holtz and Gibbs 1956, Seed et al. 1962, Jennings 1969, Chen 1975,

Johnson and Snethen 1978, Holland and Cameron 1981, Jones and Jefferson 2012) Factor Description Soil properties

Type and amount of clay minerals

The basic mineral and the fabric of particular clay, together with the salt concentration of the soil water, determine soil potential for volume change or heave.

Initial soil moisture conditions

A dry expansive soil will have higher an affinity for water and swell more than the same soil at higher water content. Conversely, a wet soil profile will lose water more readily on exposure to drying influences, and shrink more than a relatively dry initial profile.

Soil permeability The permeability of soil determines the flow rate into the soil by either gravitational flow or diffusion. Soils with higher permeability, particularly due to fissures and cracks, allow faster migration of water

CHAPTER 2 18

and promote faster rates of swell.

Plasticity Soils that exhibit plastic behavior over a wide range of moisture content and that have high liquid limit have greater potential for swelling and shrinkage.

Soil density Higher densities usually indicate closer particles spacing, which may mean greater repulsive forces between particles and larger swelling potential. Dense soils will swell more when they become wetted compared with the same soil at the same initial water content and a lower density.

Thickness of expansive strata

The thickness and location of potentially expansive soil layers in the profile considerably influence the potential movement. The greatest movement will occur when expansive soils extend down to a great depth. However, if the expansive soil is overlain by a layer of non-expansive topsoil, or overlies bedrock at a shallow depth, the movement of the soil will be greatly reduced.

Environmental conditions

Climate Climatic conditions such as precipitation and evaporation greatly influence the moisture availability and depth of seasonal moisture fluctuation in the soil profile. Expansive soils will swell and shrink if the prevailing climatic conditions lead to major moisture changes. The greatest heaves will occur under semi-arid climatic conditions that have pronounced short wet period after long dry periods.

Depth of groundwater table

Shallow and fluctuating water tables provide a source of moisture changes in the active zone near the ground surface which primarily define soil movements.

Vegetation Trees, shrubs, and grasses deplete moisture from the soil through transpiration, and have the ability to dry out soils within the zone of influence of their root systems. The drying out of soil in the vicinity of vegetation alters the pattern of soil movements by extending the drying cycle. Also, plants cause the soil to be differently wetted, the subsoil is usually subjected to nonuniform movements, and then substantial superstructure distress can take place.

Localized moisture excess

Soil heave patterns may be appreciably altered in areas where localized sources of moisture exist. Excessive garden watering and leak from broken pipe are two examples of moisture source.

Site drainage Poor surface drainage of a site leads to moisture accumulations or ponding, so significant wetting up of the soil will occur, leading to substantial heave. Improvement of site drainage can therefore dramatically reduce the magnitude of soil heave.

Surcharge pressure

The application of surcharge load on potentially expansive soil will act to balance interparticle repulsive forces and reduce soil swell. Therefore, damage due to underlying soil heave is generally associated with lightly loaded structures.

LITERATURE REVIEW 19

All of the above factors should be taken into consideration in judging the usefulness of

any method for predicting the amount and the rate of soil volume change. A summary of

the most common prediction methods are provided later in this chapter.

2.6 Soil Suction in Unsaturated Soils

Soils above the groundwater table have an affinity for water, which, either partially or

fully, fills in the space within the soil pores. This affinity that a soil has for water can be

expressed through the relative humidity of the ambient air close to the soil. This affinity

is called the total suction ψ which, by definition, is the total free energy of the soil water

determined as the ratio of the partial pressure of the water vapor in equilibrium with a

solution identical in composition to the soil water, to the partial pressure of the water

vapor in equilibrium with a pool of free pure water (Aitchison 1965). The total suction

can be calculated based on the principles of thermodynamics using Kelvin equation

(Richards 1965)

ln( )hmol

RT RV

ψ = − (2.1)

where hR is the relative humidity, which equals the ratio between the partial pressure of

pore water vapor P and the saturation pressure of water vapor over a flat surface of pure

water at the same temperature 0P , R is the universal gas constant (8.31432 J/(mol K)),

molV is the molecular volume of water vapor (0.01802 m3), and T the absolute

temperature (°K).

The total suction can also be calculated as the sum of two components; namely, the

matric suction ( )a wu u− and the osmotic suction π as shown in equation (2.2)

(Buckingham 1907, Bolt and Miller 1958, Aitchison 1965, Fredlund and Rahardjo 1993).

The matric suction component is related to the capillary phenomenon arising from the

surface tension of water (or air-water interface), and the osmotic suction component is

related to the dissolve salt in soil pore water.

CHAPTER 2 20

( )= − +a wu uψ π (2.2)

where au is the pore air pressure, and wu is the pore water pressure.

2.6.1 Matric suction

In the literature of unsaturated soils, soil suction usually refers to the matric suction

which is expressed as the excess of pore air pressure au over pore water pressure wu (i.e.,

( )a wu u− ). Matric suction has also been defined from a thermodynamics point of view as

the ratio of partial pressure of the water vapor in equilibrium with the soil water, to the

partial pressure of the water vapor in equilibrium with a solution identical in composition

with the soil water (Aitchison 1965).

It is more appropriate to consider the matric suction as a variable that expresses

quantitatively the degree of attachment of water to solid particles which results from the

general solid/water/interface interaction (Gens 2010). Because water is attracted to soil

particles and because water can develop surface tension, matric suction develops inside

the pore fluid when a saturated soil begins to dry. The action associated with matric

suction is similar to vacuum and will directly contribute to the effective stress or skeletal

forces (Mitchell 1993). Analogous to a case of the capillary tube shown in Figure (2.8),

the matric suction is related to the surface tension and the curvature of the water

meniscus and it can be calculated as

2( ) sa w

s

Tu uR

− = (2.3)

where sT is the surface tension of water, and sR is the radius of curvature. The later can

be considered analogous to the pore radius in a soil (Fredlund and Rahardjo 1993). The

smaller the pore radius of a soil, the higher the soil matric suction can be. Figure (2.9)

shows the relationship among the pore radius, matric suction, and capillary height.

LITERATURE REVIEW 21

Fig. 2.8. The capillary phenomenon contributing to the matric suction (Mitchell and Soga 2005)

Fig. 2.9. Relationship among the pore radius, matric suction, and capillary height (Fredlund and Rahardjo 1993)

Since the water acts like a membrane with negative pressure, the suction force contributes

directly to the skeletal forces like the water pressure (Figure 2.10a). As the soil continues

to dry, the water phase becomes disconnected and remains in the form of menisci or

liquid bridges at the interparticle contacts (Figure 2.10b). The curved air-water interface

produces pore water tension, which, in turn, generates interparticle compressive forces

(suction forces) that only act at particle contacts. This suction force generally depends on

CHAPTER 2 22

the separation between the two particles, the radius of liquid bridge, interfacial tension,

and contact angle (Lian et al. 1993, Mitchell and Soga 2005).

(a) (b)

Fig. 2.10. Microscopic water-soil interaction in unsaturated soils: (a) negative pore pressure acts all around the particles, (b) suction forces act only at particles contact (Mitchell and Soga 2005)

In this thesis, wu is used to represent the pore water pressure, which can be either positive

(saturated soils) or negative (unsaturated soils), and ( )−a wu u is used to represent the

matric suction, which is the absolute value of the negative pore water pressure at an

atmospheric air pressure ( 0)=au . This is consistent with the continuum mechanics

principles and has been used by several investigators (Fredlund and Rahardjo 1993).

2.6.2 Osmotic suction

Osmotic suction arises from differences in the salt concentration within the soil pore

water from that of pure water. The osmotic suction is defined by Aitchison (1965) as the

equivalent suction derived from the measurement of the partial pressure of the water

vapor in equilibrium with a solution identical in composition with the soil water, relative

to the partial pressure of water vapor in equilibrium with free pure water. The osmotic

suction can be calculated using van’t Hoff equation as

= ∆s sR T cπ (2.4)

LITERATURE REVIEW 23

where sR and sT are defined in equation (2.3), ∆c is the difference in the concentration

between two solutions.

The presence of osmotic suction gives some additional affinity for water of the soil. The

osmotic suction is related to the tendency of water to move from the region of low salt

concentration to high concentration. For example, when a pool of pure water is placed in

contact with a salt solution through a membrane, which allows only the water to flow

through, an osmotic potential will develop due to the difference in the concentration of

salt solution and water will flow through membrane. Thus, changes in osmotic suction

have no effect on the mechanical behavior (i.e., volume change and shear strength) of the

soil (Fredlund and Rahardjo 1993).

The results of the experimental research conducted by Miller and Nelson (2006) to

evaluate the effects of osmotic suction on suction measurements shows, although osmotic

suction dominates the total suction measurements, its influence on soil behavior is

relatively small compared to the effect of the changes in matric suction. Miller and

Nelson (2006) also studied the effect of salt concentration on the soil-water characteristic

curve and the soil compressibility in terms of matric suction. It was concluded that

adding salt did not result in a substantially different soil with respect to its volume change

response to changes in matric suction. Since soil osmotic suction is relatively constant at

various water contents, Krahn and Fredlund (1972) suggested that the osmotic suction

can be assumed as a constant value and subtracted from the total suction measurements.

Alonso et al. (1987) and Fredlund and Rahardjo (1993) suggested that the matric suction

is only taken into account as a relevant variable in the interpretation of unsaturated soils,

assuming that the ionic concentration of liquid in osmotic suction remains unchanged.

Hence, only matric suction has been closely related to the engineering properties of soils,

and it has been used as a tool by several researchers to estimate the properties of the soil

such as swelling behavior, shear strength, and soil compressibility (e.g., van Genuchten

1980, Alonso et al. 1990, Vanapalli et al. 1996, Rassam and Williams 1999, Rampino et

al. 2000, Vanapalli and Oh 2010).

CHAPTER 2 24

2.6.3 Matric suction profile

The potential change in matric suction is generally attributed to environmental condition,

human imposed irrigation, influence of vegetation, and accidental wetting due to broken

pipelines. Figure (2.11) shows the effect of the environmental conditions on the in-situ

profile of pore water pressure (i.e., matric suction). Dry and wet seasons cause variations

in the matric suction profile, particularly close to the ground surface. During a dry

session, the evaporation rate is high, and it results in a net loss of water from the soil. As

consequence, soil shrinkage may occur. However, if evaporation is eliminated due to any

reason such as the precipitation during wet session or covering the area, the opposite

condition may occur and soil swelling may take place (Fredlund and Rahardjo 1993). The

matric suction fluctuations are slow; however, even small suction changes may cause

significant volume changes.

The expected depth of matric suction changes in the soil profile is particularly important

as it is used to estimate soil volume change by integrating the soil strain produced over

the zone in which matric suctions change. By understanding the interaction between the

matric suction and soil volume change, a reasonable estimation of the volume change

movement of expansive soils associated with the environmental variations is possible.

Fig. 2.11. Typical pore-water pressure profiles (modified after Fredlund and Rahardjo 1993)

LITERATURE REVIEW 25

The prediction of matric suction variations is complex because of soil heterogeneity,

nonlinear unsaturated soil properties, and volume change characteristics of expansive

soils (Dye 2008). Several commercial programs, summarized in a later section, are

available for estimating the matric suction profiles considering both water flow in

unsaturated soils and soil-atmospheric interactions. Such techniques are valuable,

simple, and economical in comparison to the direct measurement of in-situ suction.

2.7 Modeling of Unsaturated Flow and Atmospheric Interactions

Water flow in the unsaturated zones is significantly influenced by the atmospheric

interactions and impacts the engineering behavior of the soil. Water flow through soils in

both saturated and unsaturated conditions can be described by Darcy’s law (Richards

1931). Darcy’s law for vertical flow is stated as follows

( )wHv ki k k h zz z

∂ ∂= − = − = − −

∂ ∂ (2.5)

where wv is the flow rate per unit area, i is the hydraulic gradient, H is the total

hydraulic head, h is the soil water pressure head, and z is the vertical distance from the

soil surface downward (i.e., the soil depth), k is the hydraulic conductivity (i.e.,

coefficient of permeability). The hydraulic conductivity for unsaturated soils is not a

constant as for saturated soils, but it is a function of soil suction (or volumetric water

content wθ ) (Figure 2.12). The capillary model shown in Figure (2.8) illustrates how

suction can be related to the effective radius of a fluid-filled pore, and helps to explain

why the relationship between hydraulic conductivity and suction is highly non-linear. As

suction increases, the largest pores drain, decreasing the open area of flow, decreasing the

effective radius of fluid filled pores, increasing the tortuosity of structure of the flow

path, and decreasing the hydraulic conductivity (see Figure 2.12).

CHAPTER 2 26

Fig. 2.12. Hydraulic conductivity as a function of soil suction (Benson 2007)

By combining the formulation of Darcy equation (2.5) with the continuity equation

w wt v zθ∂ ∂ = −∂ ∂ , and assuming isothermal conditions, isotropic hydraulic

conductivity, incompressible water phase and pore space, and static and continuous vapor

phase, the general flow equation can be written as

( ) ( )w H h kk kt z z z z z

θ∂ ∂ ∂ ∂ ∂ ∂= = −

∂ ∂ ∂ ∂ ∂ ∂ (2.6)

If soil volumetric water content wθ and pressure head h are uniquely related, then the

left-hand side of equation (2.6) can be written as w wt h h tθ θ∂ ∂ = ∂ ∂ ⋅∂ ∂ which

transforms equation (2.6) into Richards equation (Equation 2.7)

( )wh h kC kt z z z

∂ ∂ ∂ ∂= −

∂ ∂ ∂ ∂ (2.7)

where ( )w wC hθ= ∂ ∂ is the specific water capacity, which is defined as the change in

water content in a unit volume of soil per unit change in matric potential (i.e., the slope of

the relation between the volumetric water content and soil suction described through the

use of the soil-water characteristic curve (SWCC) (Figure 2.13)). The specific water

capacity varies between -∞ and 0, and can approach negative values for soils with

uniform pore size distribution (Benson 2007). The non-linearity of equation (2.7) arises

LITERATURE REVIEW 27

because the equation coefficients (k and Cw) are functions of the dependent variables, and

the exact analytical solution for specific boundary conditions is extremely difficult to

obtain. In addition, hysteresis in k and Cw further complicate the solution.

Fig. 2.13. Soil-water characteristic curve and specific water capacity (Benson 2007)

Several software packages are available for solving Richards equation (Equation 2.7).

Examples of commercially available and commonly used packages are listed in Table

(2.2). Each of these packages uses numerical methods to solve Richards equation,

considering climate boundary to simulate atmospheric interactions and root-water uptake

functions to simulate plant transpiration. Factors such as solute transport, heat transfer

and thermally driven flow, and vapor flow are also incorporated in these programs using

modified versions of equation (2.7). Each of these packages is available with a graphical

user interface and runs in the Windows (TM) operating system. The packages are similar

in conceptually as well as in functionality. However, the codes use different algorithms,

and therefore yield slightly different predictions for the same input (Benson 2007). No

definitive or universal recommendation can be provided with respect to a program that

would ensure water flow predictions are accurate (Bohnhoff et al. 2009).

CHAPTER 2 28

Table 2.2. Examples of software packages commonly used for unsaturated flow

modeling with atmospheric interactions (Benson 2007) Code Source Dimensionality Other Features HYDRUS pc-progress.com 1, 2, or 3D Solute and colloid transport, heat transfer,

dual porosity, hysteresis, snow hydrology, runoff, stochastic soil properties

SVFLUX soilvision.com 1, 2, or 3D Heat transfer, ground freezing, stochastic soil properties, runoff

UNSAT-H hydrology.pnl.gov (or) uwgeosoft.org

1D Vapor flow, heat transfer, thermally driven flow, run off, hysteresis

VADOSE/W geoslope.com 1D or 2D Oxygen transport, snow hydrology, ground freezing, run off and down slope infiltration, heat transfer, vapor flow

2.8 Volume Change Theory of Unsaturated Soils

The volume change behavior of expansive soils can be explained using the mechanics of

unsaturated soils through the use of the constitutive relationships that relate the

deformation state variables to the stress state variables. The deformation state variables

for an unsaturated soil element are the changes in total volume (i.e., soil structure) and

the changes in water volume. The volume change behavior of an unsaturated soil

primarily involves two processes; namely, the transient water flow process and the soil

volume change process. The coupled consolidation (i.e., volume change) theory of

unsaturated soils links these two processes to each other. A brief literature review of the

stress state variables and the volume change (i.e., coupled consolidation) theory of

unsaturated soils is presented in this section.

2.8.1 Stress state variables

During the early years of the development of soil mechanics, Terzaghi (1936) introduced

the concept of effective stress for saturated soils followed by other major contributions to

the definition of effective stress (Skempton 1961, Nur and Byerlee 1971). Since these

early works, the effective stress principle has been widely used in modeling geotechnical

engineering applications, giving a simple link between elastic deformation and stress

acting in the soil, the later being proportional to the observed deformation in saturated

LITERATURE REVIEW 29

soils (Nuth and Laloui 2008). The effective stress ′σ is a combination of both the

externally applied stress σ and the internal pressure of pore water wu , and it is expressed

as

wuσ σ′ = − (2.8)

The use of Terzaghi’s effective stress has been well accepted and experimentally verified

for saturated soils (Rendulic 1936, Bishop and Eldin 1950, Laughton 1955, Skempton

1961). The success associated with the use of the Terzaghi’s effective stress principle for

saturated soils has prompted early investigators to extend the effective stress principle to

unsaturated soils and provide unified formulations of the effective stress as shown in

Table (2.3).

Table 2.3. Effective stress equations for unsaturated soils (modified after Lu 2010) Equation Notations Reference

σ σ′ ′′= + p p′′ : pore-water pressure deficiency Donald (1956)

( )a cu uσ σ′ = − + cu : capillary pressure Hilf (1956)

σ σ β′ ′= − wu β ′ : holding or bonding factor which is a measure of the number of bonds under tension effective in contributing to shear strength of the soil

Croney et al. (1958)

( ) ( )a a wu u uσ σ χ′ = − + − χ : effective stress parameter related to the soil degree of saturation

Bishop (1959)

pσ σ ϕ′ ′′= − ϕ : parameter varying between zero to one depending on the degree of saturation

Aitchison (1960)

pσ σ β′ ′′= − β : statistical factor of the same type as the contact area

Jennings (1961)

( ) ( )i a w au u uσ σ χ σ′ = − − + − iσ : intrinsic stress arising from inter-particle forces

Newland (1965)

( )( )

a m m a

s s a

u h uh u

σ σ χχ′ = − + +

+ + mh : matric suction

sh : solute suction

mχ : effective stress parameter for matric suction

sχ : effective stress parameter for solute suction

Richards (1965)

CHAPTER 2 30

m m s sp pσ σ χ χ′ ′′ ′′= + + mp′′ : matric suction

sp′′ : solute suction

mχ , sχ : soil parameters which are dependent upon the stress path

Aitchison (1973)

( ) ( )a w a wu a u uσ σ′ = − + − wa : ratio of area of water-mineral and water-water contact of total area of “wavy” plane

Lambe and Whitman (1979)

( ) ( )a a w su u u R Tσ σ χ ς′ = − + − − −

χ : parameter representing the proportion of the total void area occupied the water on a reference plane R : osmotic suction ς : interface parameter on the reference plane

Allam and Sridharan (1987)

0.55

( ) ( ),

( ):

( )

a a w

a w

a w b

u u u

u uwhere

u u

σ σ χ

χ−

′ = − + −

−= −

χ : effective stress parameter ( )a w bu u− : air entry value

Khalili and Khabbaz (1998)

* ( (1 ) )ij ij w a ijS u S uσ σ δ= − + − ijσ : total stress tensor

ijδ : Kroneker delta or substitution tensor *ijσ : Bishop’s average soil skeleton stress

S : degree of saturation

Jommi (2000)

The need for a unified formulation of the stress variables rose from the complexity of

modeling the behavior of porous materials (saturated and unsaturated soils). The single-

valued effective stress principle converts the analysis of a multiphase porous media into a

mechanically equivalent, single-phase, single-stress state continuum comprehension

(Nuth and Laloui 2008). As noted by Khalili and Khabbaz (1998), the advantage of the

effective stress approach is that the change in the shear strength with changes in total

stress, pore water pressure, and pore air pressure can be related to a single stress variable.

As a result, a complete characterization of the soil strength requires matching of a single

stress history rather than two or three independent stress variables. Furthermore, the

approach requires very limited testing of soils in an unsaturated state.

However, limitations of the single-valued effective stress principle have been cited in the

literature by many researchers (Coleman 1962, Aitchison 1965, Blight 1965, Burland

1965, Matyas and Radhakrishna 1968, Barden et al. 1969, Brackley 1971, Fredlund and

LITERATURE REVIEW 31

Morgenstern 1977, Gens et al. 1995). All the formulations shown in Table (2.3)

incorporate a soil parameter in order to form a single valued effective stress variable.

Jennings and Burland (1962) and many subsequent authors have shown that whatever the

relationship chosen for soil parameter, there is no unique relationship between volumetric

strain and effective stress, and that single-valued effective stress principle is not valid.

Jennings and Burland (1962) stated that the volume change and shear strength of

unsaturated soils cannot be related to a single effective stress. Rather, more than one

stress state variable should be used to describe the behavior of unsaturated soils. Fredlund

and Morgenstern (1977) and Fung (1977) further argued that the variables used for the

description of a stress should be independent of the material properties. Due to the

experimental difficulties to evaluate the soil parameter, and the philosophical difficulties

to justify the use of soil properties in the description of a stress state, the single-valued

effective stress equations have not received much attention in describing the mechanical

behavior of unsaturated soils. However, in another recent viewpoint, Khalili et al. (2004)

criticized the theoretical basis for this argument. They pointed out that in a multiphase

porous medium, such as saturated and unsaturated soils, the stress state within each phase

will naturally be a function of the properties of that phase as well as the other phases

within the system. This is required in order to ensure deformation compatibility between

the phases. In addition, Nuth and Laloui (2008) suggested that soil parameter might also

be related to the current stress and stress history; hence, a unique relationship between the

soil parameter and the ratio of matric suction over the air entry value can be obtained for

most soils.

Fredlund and Morgenstern (1977) proposed two independent stress variables to describe

the constitutive behavior of unsaturated soils. The identification of state variables can be

based on multiple continuum mechanics, leading to the conclusion that any two of the

three possible state variables (σ , wu , au ) can be used to define the stress state (Fredlund

and Morgenstern 1977), possible combinations being:

( )auσ − and ( )a wu u− , e.g., used in Alonso et al. (1990) (2.9a)

( )− wuσ and ( )a wu u− , e.g., used in Geiser et al. (2000) (2.9b)

CHAPTER 2 32

( )auσ − and ( )− wuσ (2.9c)

For physical and practical reasons, the most frequently used stress variables in the two

independent stress state variables approach are the net normal stress ( )auσ − and the

matric suction ( )a wu u− (Equation 2.9a) (e.g., Matyas and Radhakrishna 1968, Alonso et

al. 1990). The matric suction ( )a wu u− has a definite physical meaning, while most of the

time the air pressure may be considered constant and equal to the atmospheric pressure

( 0)a atmu u= = . Under this assumption, the net normal stress is simplified to the total

normal stress and the matric suction is equal to the absolute value of the negative pore

water pressure. Moreover, this choice is adapted to the axis translation technique,

consisting in the application of ( 0)>au (Nuth and Laloui 2008). Fredlund and

Morgenstern (1977) experimentally validated the principle of independent stress state

variables for unsaturated soils using null tests. The components of the proposed stress

state variables (σ , au , wu ) were varied equally in order to maintain constant values for

stress state variables (i.e., ( )auσ − and ( )a wu u− ). As there are no overall changes in the

state of the soil due to changing in the components of the stress state variables, the stress

state variables are valid well for unsaturated soils. Tarantino et al. (2000) investigated

these stress state variables using a new laboratory apparatus designed and constructed to

test unsaturated soils in a broad range of degree of saturation and negative pore water

pressures. The results confirmed the use of the net normal stress and matric suction as

stress state variables.

Re-examination of the independent stress state variables proposed by Fredlund and

Morgenstern (1977) has led several ongoing attempts to develop modified stress variables

to describe the behavior of unsaturated soils (e.g., Alonso et al. 1990, Kohgo et al. 1993,

Kato et al. 1995, Wheeler and Sivakumar 1995, Wheeler et al. 2003, Blatz and Graham

2003, Gallipoli et al. 2003). There are several state-of-the-art reports and review papers

over the last years; for examples, Gens (1996), Wheeler and Karube (1996), Kohgo

(2003), Gens et al. (2006), Sheng and Fredlund (2008), Sheng et al. (2008), Gens (2009),

Cui and Sun (2009), Gens (2010), and Sheng (2011). These papers may serve as good

references for studying the alternative stress state variables or constitutive variables that

LITERATURE REVIEW 33

can be used to establish various models for unsaturated soils. The recent proposed stress

variables are clearly more complex than the traditional variables of net normal stress and

matric suction.

Wheeler and Karube (1996) discussed the justification of using the more complicated

stress variables and discussed some disadvantages that might arise from using that

approach. First, it would be more difficult for practicing engineers to think in terms of the

new stress variables even when describing a relatively simple stress path, e.g.,

drying/wetting under constant applied load. Second, it would be more difficult to devise

simple experiments to obtain the model parameters. Only if the new stress variables have

a strong physical significance, resulting in considerable improvement in modeling

capacity and more simplicity in stress-strain relationships, the use of new stress variables

would be justified (Jotisankasa 2005).

2.8.2 Volume change constitutive relationships

In developing the analysis of volume change behavior of unsaturated expansive soils, it is

necessary to express soil volume change in terms of stress state variables and appropriate

strain variables through specified constitutive relations. Different constitutive

relationships have been proposed to interpret the volume change behavior of the soil.

Several researchers (Biot 1941, Coleman 1962, Matyas and Radhakrisha 1968, Barden et

al. 1969, Aitchison and Woodburn 1969, Brackley 1971, Aitchison and Martin 1973,

Fredlund and Morgenstern 1976, 1977, Fredlund and Rahardjo 1993, Zhang 2004)

developed volume change constitutive relationships based on the assumption that the soil

is elastic in nature for a large range of loading conditions. Two constitutive relationships

have been suggested for describing the deformation state of unsaturated soil. One

constitutive relationship is formulated for soil structure (in terms of void ratio or

volumetric strain) and the other constitutive relationship is formulated for water phase (in

terms of degree of saturation or water content). Two independent stress variables (i.e., net

normal stress and matric suction) are used in the formulations. In total four volumetric

deformation coefficients are required to link the stress and deformation states. The

constitutive relations can be graphically presented in the form of a deformation state

variable versus two independent stress state variables, and they can be formulated in

CHAPTER 2 34

different forms, namely soil mechanics formulation, compressibility formulation, and

elasticity formulation (Fredlund et al. 2012).

In another viewpoint, a variety of elasto-plastic constitutive relationships or models has

been introduced and studied (e.g., Alonso et al. 1987, Karube 1988, Alonso et al. 1990,

Gens and Alonso 1992, Kohgo et al. 1993, Modaressi and Abou-Beker 1994, Bolzon et

al. 1996, Cui et al. 1995, Delage and Graham 1995, Kato et al. 1995, Wheeler and

Sivalumar 1995, Wheeler et al. 2003, Blatz and Graham 2003, Chiu and Ng 2003,

Tamagnini 2004, Thu et al. 2007, Sheng et al. 2008). Lloret and Alonso (1980)

established that the constitutive models based on the concept of elastoplasticity provide a

better understanding and explanation of expansive soil behavior, on particular, those

features concerning stress path dependency and soil collapse upon wetting. However,

Alonso et al. (1990) argued that, for pavements or shallow foundations built on expansive

soils, the assumption that expansive soils are elastic is considered as a reasonable

assumption where drying–wetting process can cause an unsaturated expansive soil to

yield. Similar arguments were also put forward by Zhang and Briaud (2010) since it is

reasonable to assume the soil has experienced the maximum wetness and dryness in the

past. In other words, if an expansive soil has some plasticity, this kind of plasticity could

have been eliminated by a long history of wetting-drying cycles. This may be also the

reason why most expansive soils are usually heavily overconsolidated. Since the volume

of the expansive soil is influenced by the mechanical stress, one may argue that the soil

will yield under a combination of mechanical stress and matric suction variations.

However, for pavements and light residential or commercial buildings where the majority

of the soil volume change problems are likely to occur, the mechanical stress due to

repeated traffic or superstructure load is very small that it will not cause soil yielding. As

a result, for engineering practice applications, expansive soils can be assumed to be

elastic (Zhang and Briaud 2010). Consequently, the details of elasto-plastic constitutive

relationships will not be covered here. Only Fredlund and Morgenstern (1976, 1977)

constitutive relationships for volume change behavior that are employed in the thesis will

be discussed in detail in the following.

LITERATURE REVIEW 35

The unsaturated soil is considered as a four-phase mixture (Fredlund 1979), with two

phases that come to equilibrium under applied stress (i.e., soil particle and contractile

skin) and two phases that flow under applied pressure (i.e., air and water). The total

volume change of a soil element must be equal to the sum of volume changes associated

with each phase. If the soil particles are assumed incompressible and the volume change

of the contractile skin are assumed internal to the element, the continuity requirement for

an element of unsaturated soil is (Fredlund and Morgenstern 1976)

0 0 0

v w aV V VV V V∆ ∆ ∆

= + (2.10)

where 0V is initial overall volume of an unsaturated soil element, vV is volume of soil

voids, wV is volume of water in the soil element, and aV is volume of air in the soil

element.

The above continuity requirement shows, in order to describe the volume change

behavior in an unsaturated soil, the volume changes associated with any two of the three

above volume variables must be measured or predicted, while the third volume change

can be computed. In practice, the overall volume change 0( )vV V∆ and the water volume

change 0( )wV V∆ are usually measured, while the air volume change 0( )aV V∆ is

calculated as the difference between the volume change of soil structure and water phase

(Fredlund and Rahradjo 1993).

By assuming the soil behaves as an incrementally isotropic, linear elastic material, the

soil structure constitutive relations associated with the strains can be written as

( ) ( )( ) ( )x a a wx y a z a

d u d u ud d u d uE E H

σ µε σ σ− − = − − + − + (2.11a)

[ ]( ) ( )( ) ( )y a a w

y x a z a

d u d u ud d u d uE E H

σ µε σ σ− −

= − − + − + (2.11b)

( ) ( )( ) ( )z a a wz x a y a

d u d u ud d u d uE E H

σ µε σ σ− − = − − + − + (2.11c)

CHAPTER 2 36

, ,yz xyzxyz zx xy

d ddd d dG G Gτ ττγ γ γ= = = (2.11d)

where xσ , yσ , and zσ are normal stresses in the x-, y-, and z-directions, respectively, xε ,

yε , and zε are normal strains in the x-, y-, and z-directions, respectively, ( )x auσ − ,

( )y auσ − , and ( )z auσ − are net normal stresses in the x-, y-, and z-directions,

respectively, µ is Poisson’s ratio, E is modulus of elasticity for the soil structure with

respect to a change in net normal stress, H is modulus of elasticity for the soil structure

with respect to a change in matric suction, yzγ , zxγ , and γ xy are shear strains on the x-, y-,

and z-planes, respectively, and G is shear modulus.

The deformation variable associated with the overall or total volume change 0( )vdV V

can be written as the sum of the normal strains (Equation 2.12).

0

1 2 33( ) ( ) ( )−= = + + = − + −v

v x y z mean a a wdV d d d d d u d u uV E H

µε ε ε ε σ (2.12)

where vε is volumetric strain, meanσ is mean total normal stress ( meanσ =

( ) 3x y zσ σ σ+ + ), in which xσ , yσ , and zσ are normal stresses in the x-, y-, and z-

directions, respectively. The total volume change refers to the volume change of the soil

structure, and equation (2.12) is the constitutive equation for soil structure.

The constitutive equation for the water phase defines the water volume change in the soil

element for any change in the total stress and matric suction. By assuming water is

incompressible, the constitutive equation for the water phase can be formulated as a

linear combination of the stress state variables changes as below (Fredlund and Rahardjo

1993).

0

( )( ) ( ) ( )

3 1( ) ( ) (2.13)

y aw x a z a a w

w w w w

mean a a ww w

d udV d u d u d u uV E E E H

d u d u uE H

σσ σ

σ

−− − −= + + +

= − + −

LITERATURE REVIEW 37

where wE is water volumetric modulus associated with a change in net normal stress, and

wH is water volumetric modulus associated with a change in matric suction.

Fredlund and Morgenstern (1976) proposed the following constitutive relationships for

volume change of soil structure and water phase in a compressibility form

1 20

( ) ( )= = − + −s svv mean a a w

dVd m d u m d u uV

ε σ (2.14)

1 20

( ) ( )w wwmean a a w

dVd m d u m d u uV

θ σ= = − + − (2.15)

where 1sm is coefficient of total volume change with respect to change in net normal

stress, 2sm is coefficient of total volume change with respect to change in matric suction,

1wm is coefficient of water volume change with respect to change in net normal stress,

and 2wm is coefficient of water volume change with respect to change in matric suction.

Comparing equations (2.14) and (2.15) with (2.12) and (2.13), the volume change

coefficients can be related to the elastic moduli E and H , volumetric modulus wE and

wH , and Poisson’s ratio µ as follows

11 23( )sm

Eµ−

= , 23smH

= , 13w

w

mE

= , and 21w

w

mH

= (2.16)

The constitutive relationships for soil structure and water phase of an unsaturated soil can

be presented graphically in the form of constitutive surfaces (Figure 2.14). The

deformation state is plotted with respect to the stress state variables ( )mean auσ − and

( )a wu u− . All the volume change coefficients in equations (2.14) and (2.15) can be

determined from the constitutive surfaces as shown in Figure (2.14), which are the slopes

of the constitutive surface at a point.

CHAPTER 2 38

(a)

(b)

Fig. 2.14. Three-dimensional constitutive surfaces for unsaturated soil: (a) soil structure constitutive surface, (b) water phase constitutive surface (modified after Fredlund et al. 2012)

Conventional soil mechanics terminology makes the use of void ratio e, gravimetric

water content w, and degree of saturation S to define the volume-mass properties of

unsaturated soils. Therefore, the constitutive equations for unsaturated soils can be

LITERATURE REVIEW 39

written in terms of void ratio and gravimetric water content as the deformation state

variables for soil structure and water phase, respectively.

( ) ( )t mean a m a wde a d u a d u uσ= − + − (2.17)

( ) ( )t mean a m a wdw b d u b d u uσ= − + − (2.18)

where ta is coefficient of compressibility with respect to change in net normal stress, ma

is coefficient of compressibility with respect to change in matric suction, tb is coefficient

of water content change with respect to a change in net normal stress, and mb is

coefficient of water content change with respect to a change in matric suction. Equations

(2.17) and (2.18) can also be visualized as constitutive surface on a three-dimensional

plot. Each abscissa represents one of the stress state variables and the ordinate represents

the soil volume-change properties (Figure 2.15). The compressibility coefficients

, , ,t m ta a b and mb are another form of the volume change coefficients, which can be

determined as the slopes of the void ratio and water content constitutive surfaces at a

point as shown in Figure (2.15).

The volume change coefficients of equations (2.14) and (2.15) ( 1 2 1 2, , ,s s w wm m m m ) can

be expressed in terms of the coefficients of equations (2.17) and (2.18) ( , , ,t m t ma a b b ),

and then the volume change coefficients of soil can be obtained from void ratio and water

content constitutive surfaces (Figure 2.15) (Fredlund and Rahardjo 1993) as follows

10 0

1 11 ( ) 1

st

mean a

dem ae d u eσ

= =+ − +

(2.19)

20 0

1 11 ( ) 1

sm

a w

dem ae d u u e

= =+ − +

(2.20)

10 01 ( ) 1

w s st

mean a

G Gdwm be d u eσ

= =+ − +

(2.21)

20 01 ( ) 1

w s sm

a w

G Gdwm be d u u e

= =+ − +

(2.22)

CHAPTER 2 40

where 0e is initial void ratio prior to deformation, and sG is the specific gravity of the soil

solids.

(a)

(b)

Fig. 2.15. Three-dimensional constitutive surfaces for unsaturated soil expressed using soil mechanics terminology: (a) void ratio constitutive surface, (b) water content constitutive surface (modified after Fredlund et al. 2012)

LITERATURE REVIEW 41

The volume change coefficients can, in general, be obtained from the consolidation tests

or triaxial tests with suction control. However, such tests are usually time consuming,

costly, and may not be reasonable in engineering practice (Fredlund and Raharajo 1993).

Vu and Fredlund (2006) proposed a method to calculate the four coefficients of soil

volume change. The void ratio constitutive surface of unsaturated soil is estimated in

terms of the compressive indices obtained from the conventional oedometer tests.

However, this method estimates unreasonably large soil deformations at low net normal

stresses and/or low suctions. More details about Vu and Fredlund (2006)’s method are

provided in a later section.

2.8.3 Coupled consolidation theory for unsaturated soils

The rigorous formulation for consolidation (i.e., volume change) of unsaturated soils

requires that the continuity equation be coupled with the equilibrium equations (Fredlund

and Hasan 1979, Dakshanamurthy and Fredlund 1980, Lloret and Alonso 1980,

Dakshanamurthy et al. 1984, Lloret et al. 1987, Fredlund and Rahardjo 1993, Wong et al.

1998, Vu 2002). In a three-dimensional consolidation problem, there are five unknowns of

deformation and volumetric variables to be solved. These unknowns are the displacements

in the x-, y-, and z-directions and the water volume change and air volume change. The

displacements in the x-, y-, and z-directions are used to compute the total volume change.

The five unknowns can be obtained from three equilibrium equations for the soil structure

and two continuity equations (water and air phase continuities). These equations require

constitutive relations for the volume change of unsaturated soils as well as flow laws for

fluid phases (air and water phases). However, the pore air pressure is generally assumed to

be atmospheric and remains unchanged during the consolidation process. In this case, only

stress equilibrium condition and water flow continuity need to be considered in the

analysis.

2.8.3.1 Equilibrium equations for soil structure

The stress state for an unsaturated soil element should satisfy the following equilibrium

conditions

, 0ij j jbσ + = (2.23)

CHAPTER 2 42

where ,ij jσ are components of the net total stress tensor, and jb are components of body

force vector.

The strain-displacement equations (Cauchy’s Equation) for soil structure of an

unsaturated soil are given as follows

, ,1 ( )2ij i j j iu uε = + (2.24)

where ijε are components of the strain tensor, and iu are components of displacement in

the i-direction.

By substituting the strain-displacement equation (Equation 2.24) and the stress-strain

relationship (Equation 2.12) into the equilibrium equation (Equation 2.23), the

differential equations for soil structure for general three-dimensional problems can be

written as

2 ( )1( ) (3 2 ) 0v a wx

u uG G u G bx H xελ λ∂ ∂ −

+ + ∇ − + + =∂ ∂

(2.25a)

2 ( )1( ) (3 2 ) 0v a wy

u uG G v G by H yελ λ∂ ∂ −

+ + ∇ − + + =∂ ∂

(2.25b)

2 ( )1( ) (3 2 ) 0v a wz

u uG G w G bz H zελ λ∂ ∂ −

+ + ∇ − + + =∂ ∂

(2.25c)

where [ ](1 )(1 2 )Eλ µ µ µ= + − , u, v, and w are displacements in the x-, y-, and z-

directions, respectively, and xb , yb , and zb are body force in the x-, y-, and z-directions,

respectively.

2.8.3.2 Water continuity equation

The water continuity equation for unsaturated soils, assuming that water is

incompressible and deformations are incrementally infinitesimal, can be written as

(Freeze and Cherry 1979)

LITERATURE REVIEW 43

0( / ) x y zw w w wV V v v vi j k

t x y z∂ ∂ ∂ ∂

= + +∂ ∂ ∂ ∂

(2.26)

where 0( )wV V t∂ ∂ is net flux of water per unit volume of the soil, t is time, and

x y zw w w wv v i v j v k= + + is Darcy’s flux which relates to the hydraulic head (i.e., pressure

head plus elevation head) using Darcy’s law

wwi wi

i w

uv k Yx gρ

∂= − + ∂

(2.27)

where wiv is Darcy’s flux in the i direction, wik is hydraulic conductivity in the i direction

which is a function of matric suction, wu is pore water pressure, wρ is density of water, g

is gravitational acceleration, and Y is elevation.

Fredlund and Rahardjo (1993) derived the differential equation for water phase (Equation

2.28) by substituting the time derivative of the water phase constitutive equation

(Equation 2.13 or 2.15) and Darcy’s law (Equation 2.27) into the water phase continuity

equation (Equation 2.26).

1 2( ) ( )w w x ymean a a w w w

w ww w

u u u u um m k Y k Yt t x x g y y g

σρ ρ

∂ − ∂ − ∂ ∂ ∂ ∂+ = + + + ∂ ∂ ∂ ∂ ∂ ∂

z ww

w

uk Yz z gρ

∂ ∂+ + ∂ ∂

(2.28)

Fredlund and Rahardjo (1993) further derived equation (2.28) by extending Biot’s

consolidation theory for saturated soils (Biot 1941). Equation (2.12) (or 2.14) was solved

for ( )mean ad uσ − in terms of vdε and ( )a wd u u− , and then ( )mean ad uσ − was substituted

into equation 2.13 (or 2.15). The volumetric water content variations can be expressed as

1 20

( )ww v w a w

dVd d d u uV

θ β ε β= = + − (2.29)

CHAPTER 2 44

where 11

1

w

w s

mm

β = , and 1 22 2

1

β = −w s

ww s

m mmm

.

By substituting equation (2.29) into the left-handed side of equation (2.28), the

differential equation for water phase can be obtained as

1 2( )v a w

w wu u

t tεβ β∂ ∂ −

+ =∂ ∂

x yw ww w

w w

u uk Y k Yx x g y y gρ ρ

∂ ∂ ∂ ∂+ + + ∂ ∂ ∂ ∂

z ww

w

uk Yz z gρ

∂ ∂+ + ∂ ∂

(2.30)

Equations (2.25) and (2.30) together are the differential equations for the coupled

consolidation for unsaturated soils that can be used to predict the volume change

behavior of unsaturated soils (Fredlund and Rahardjo 1993).

2.9 Volume Change Predictions

The uncertainty in the estimation of volume change behavior of expansive soils can be of

concern for geotechnical engineering practitioners as it may contribute to several

undesirable outcomes: (i) expensive foundation systems due to overestimation of volume

change; (ii) litigation due to underestimation of volume change; (iii) growth in a number

of local protocols that have limited applicability; and (iv) lack of confidence in the future

performance of existing and newly designed structures on expansive soils (Singhal 2010).

Hence, it is important to provide tools for practitioners to reliably estimate the volume

change behavior of expansive soils in the field. Significant advances were made during

the last half-a-century towards prediction of the heave and the shrink related volume

change behavior of expansive soils. This section presents a critical review of the state-of-

the-art of methods for predicting the volume change movement of expansive soils.

2.9.1 Methods for predicting heave potential

The focus of most prediction methods proposed in the literature has been towards

estimating the swelling characteristics; namely, heave/swell potential and swelling

LITERATURE REVIEW 45

pressure. The heave potential is defined as the ratio of increase in thickness H∆ to the

original thickness H of a laterally confined sample on soaking under 7 kPa surcharge,

after being compacted to the maximum density at the optimum water content in the

standard AASHTO compaction test (Seed et al. 1962). However, the swelling pressure is

the pressure required to hold the soil, or restore the soil, to its initial void ratio when

given access to water (Shuai 1996). The estimation of the swelling pressure was beyond

the scope of this study, thus only methods for predicting the heave potential are briefly

described in this section. The available methods can be categorised into: (i) oedometer

methods, (ii) empirical methods, and (iii) suction-based methods.

2.9.1.1 Oedometer methods

Considerable research has been conducted to predict the soil heave potential based on the

results of oedometer tests (e.g., Jennings and Knight 1957, Salas and Serratosa 1957,

Lambe and Whitman 1959, Clisby 1963, Sullivan and McClelland 1969, Aitchison et al.

1973, Smith 1973, Fredlund et al. 1980, Weston 1980, Justo and Saetersdal 1981,

Dhowian 1990, Nelson and Miller 1992, Abdullah 2002, Nelson et al. 2006, Nelson et al.

2012). The oedometer based methods require representative undisturbed samples

collected from the active zone depth typically in a dry season. The samples are then

restrained laterally and loaded axially in a consolidometer with access to free water to

saturation. The magnitude of the heave potential in oedometer testing can be estimated by

applying the consolidation theory in reverse (Wanyan et al. 2008)

0

01fe eH

H e−∆

=+

(2.31)

where H∆ is soil heave, H is soil layer thickness, fe and 0e are initial and final void

ratios, respectively.

In oedometer tests, it is vital to follow as closely as possible the expected stress sequence

to which the soils will be subjected in the field. Therefore, there are different opinions in

the published methods concerning the simulation of field conditions in the oedometer

tests (Dhowian 1990). A list of various methods utilizing the oedometer test results in

CHAPTER 2 46

estimating the heave potential is presented in Vanapalli and Lu (2012). The most

common oedometer methods along with the interpretation of the actual stress path that is

being followed in each method are reviewed here.

Direct method (Texas Highway Department Method TEX-124-E)

The direct method is based on a free swell oedometer test conducted on an undisturbed

specimen to model the field behavior with a zero additional applied surface loading

(Figure 2.16). In the free swell test, undisturbed specimens of the soils are inundated

while only a token load (seating pressure = 1 to 7 kPa) is applied and vertical

deformations are recorded. The common modification to the free swell test is to apply the

field overburden plus structural load stress, the specimen and then to inundate and

observe swell. The stress path followed when using this test procedure is shown in Figure

(2.16).

Fig. 2.16. Stress path followed when using the direct method (Fredlund et al. 1980)

NAVFAC (1971) outlined the procedure of the direct method for estimating the

magnitude of heave that may occur when footings are built on expansive soils.

Undisturbed, unsaturated soil specimens are extracted from the subsoil at different

elevations up to a depth of zero swell. Each specimen is subjected to a free swell test with

the sum of the field overburden pressure and the anticipated structural load that is applied

as surcharge load on the specimen. The test results are plotted as percent swell versus

LITERATURE REVIEW 47

depth as shown in Figure (2.17a). The area under the percent swell versus depth curve,

integrated upward from the depth of zero swell, represents the total heave/swell. This

total swell is plotted versus depth to predict the swell at any depth (Figure 2.17b). Smith

(1973) presented a similar procedure, and provided an example to illustrate how the

direct method works for the determination of heave potential in soil strata. This is

valuable in deciding the methods of construction to be employed and the remedial

procedures to use in securing the greatest value for construction money.

(a) (b)

Fig. 2.17. Heave calculations using the direct method (NAVFAC 1971)

The greatest virtue of the direct method is its simplicity yet applicability to the field

condition. However, Fredlund et al. (1980) found that the predicted heave is significantly

below the actual heave experienced in the field. It was anticipated that the

underestimation of the amount of field heave would be primarily due to a lack of

accounting for sampling disturbance and a great difficulty in securing full water-uptake in

the oedometer specimen. Abdullah (2002) experimentally showed that the direct method

overestimates the field heave because the oedometer test allows simulation of soil heave

in the vertical direction only and does not account for the reduction in the vertical heave

CHAPTER 2 48

due to the lateral soil swelling in the field. Abdullah (2002) introduced a heave reduction

factor to adjust the predicted heave in order to represent the real in situ vertical heave.

Sullivan and McClelland (1969) method

Sullivan and McClelland (1969) proposed a heave prediction method based on constant

volume oedometer tests on undisturbed specimens. The undisturbed specimen is initially

subjected to in situ overburden pressure and allowed to come to equilibrium (Figure

2.18). The specimen is then allowed free access to water and maintained at constant

volume by adding loads until no more swelling tendency is observed (i.e., the swelling

pressure is reached). The sample is then unloaded and allowed to swell by decreasing the

loads in small increments.

The method can be used to estimate the soil heave occurred due to reduction in

overburden pressure (unloading). However, Fredlund et al. (1980) mentioned that this

method is expected to underestimate the actual heave if the sampling disturbance will not

be taken into consideration. It was suggested that the results of constant volume

oedometer tests should be adjusted for the effects of compressibility of the apparatus

prior to their interpretation. Figure (2.19) shows the manner in which an adjustment

should be applied to the laboratory data. The (uncorrected) swelling pressure must also be

corrected for sampling disturbance as shown in Figure (2.20). The correction procedure

was similar to the Casagrande’s construction used for determining the preconsolidation

pressure of a saturated soil. Details of the correction procedure are explained in Fredlund

and Rahardjo (1993).

The constant volume oedometer test method has been used widely and considered to be

one of the most reliable methods for the determination of swell characteristics (i.e., heave

potential and swell pressure). Meanwhile, it is one of the most difficult and cumbersome

methods of measuring the swell characteristics. This is probably due to the difficult and

somewhat impossible restrictions for the constant volume test such as controlling the

vertical deformation by 0.005-0.01 mm, which requires computer control and also careful

adjustments for apparatus compliance (Abbaszadeh 2011). Also, it could be due to the

LITERATURE REVIEW 49

difficulty to secure water entry when the specimen is under high applied loadings which

are necessary for the test (Jennings 1969).

Fig. 2.18. Stress path followed when using Sullivan and McClelland method (Fredlund et al. 1980)

Fig. 2.19. Adjustment of laboratory test data to compensate for compressibility of oedometer apparatus (Fredlund and Rahardjo 1993)

CHAPTER 2 50

Fig. 2.20. Construction procedure to correct for sampling disturbance (Fredlund and Rahardjo 1993)

Double-oedometer method

Jennings and Knight (1957) proposed the double odometer method based on the results of

two oedometer tests, namely, the free-swell oedometer test and the natural water content

oedometer test. The two tests were conducted, as explained in the direct method, on

identical specimens initially subjected to a token load of 1 kPa. However, no water is

added to the oedometer pot during the natural water content test. To compute the soil

heave, the data of the natural water content oedometer test are adjusted vertically to

match the results of free swell test at high applied loads. The stress paths followed by the

double odometer tests in terms of net normal stress and matric suction are shown in

Figure (2.21).

Jennings and Knight (1957) and other researchers (Weston 1980, Justo and Saetersdal

1981, Abdullah 2002) found that the double odometer method overestimates the actual

heave. This overestimation appears to be primarily due to the dependency of the method

on the one-dimensional oedometer tests which assume the heave potential is manifested

only in the vertical direction. In addition, the specimens in the double oedometer tests are

allowed to swell under the token load only, thus, producing a higher value of soil heave

LITERATURE REVIEW 51

(Abdullah 2002). Another drawback is that the difficulty to obtain identical undisturbed

specimens in order to conduct the odometer tests. However, Fredlund et al. (1980)

suggested that the heave prediction using the double oedometer method is generally

satisfactory since the data analysis takes into consideration the effect of sample

disturbance.

Fig. 2.21. Stress paths followed when using double oedometer method (Jennings and Knight 1957)

Nelson and Miller (1992) method

Nelson and Miller (1992) method is based on data obtained from the overburden swell

test and the constant volume oedometer test. The two tests are conducted on identical

samples obtained from the same depth. Typical test results for both types of oedometer

tests are shown in Figure (2.22). In the overburden swell test, the percent swell S (%)

corresponds to the particular value of the vertical stress applied at the time of inundation 'iσ for the conditions under which heave is being computed. In the constant-volume test,

the percent swell is zero at an inundation pressure of 'cvσ . Thus, points B and D, as

shown in Figure (2.22), fall on the line representing the desired relationship between 'iσ

and S (%). This relationship is a straight line (i.e., heave line BD) on a semi-logarithmic

CHAPTER 2 52

plot. At any depth in the soil, the percent swell %S will fall along the straight line BD.

The slope of that line is defined as the heave index HC and is given by

'

'

(%)

log σσ

=

Hcv

i

SC (2.32)

where HC is heave index, %S is percent swell, 'cvσ is swelling pressure from the

constant-volumetric oedometer test, and ′iσ is inundation stress subjected to a sample in

the overburden swell test which equals to the overburden stress in the field for the

conditions under which heave is being computed. If values of HC and 'cvσ are known,

the vertical strain or percent swell that will occur during inundation at any depth z in a

soil profile can be determined from equation (2.32). For the case of free field heave,

when the soil at a depth z is inundated, the stress on the soil is the overburden

stress '( )ob zσ . This value is, therefore, the inundation stress 'iσ in the field and equation

(2.32) can be rewritten as equation (2.33).

( ) (%) log( )( )

σεσ

′= =

′cv

v z z Hob z

S C (2.33)

For a layer of soil of thickness iz∆ that exists at a depth z to its midpoint, the maximum

heave iρ , that will occur due to the expansion of that layer during a complete inundation,

can be obtained by multiplying the vertical strain ( )v zε (Equation 2.33) by the layer

thickness iz∆ ; thus,

log( )( )

cvi H i

ob z

C z σρσ

′= ∆

′ (2.34)

In the actual application of equation (2.34), a soil profile will be divided into layers of

thickness iz∆ , the value of heave for each layer will be computed, and the incremental

values will be added to determine the total heave in the field (Nelson et al. 2011).

LITERATURE REVIEW 53

Fig. 2.22. Hypothetical oedometer test results (modified after Nelson and Miller 1992)

Nelson and Miller method utilizes the mechanical stress as the controlling stress state

variable to calculate the heave potential, and gives conservative estimates of soil heave

(Nelson and Miller 1992). However, it has been indicated that the determination of the

heave index HC by conducting the overburden swell test and the constant volume

oedometer test on identical samples is generally not practical, mainly because it is almost

impossible to obtain two identical samples from the field (Nelson et al. 2012). To

accurately determine HC , Nelson et al. (2006) suggested that several overburden swell

tests at different inundation pressures along with a constant volume oedometer test would

be required. This is also neither practical nor economical for engineering practice;

therefore, a relationship between the swell pressure from the constant volume oedometer

test ′cvσ and the swell pressure from the overburden swell test ′csσ is proposed so that the

value of the heave index HC can be determined from a single overburden swell test

(Nelson et al. 2006). The relationship is represented as

( )′ ′ ′ ′= + −cv i cs iσ σ λ σ σ (2.35)

where λ is a parameter. The rationale behind this equation is that the value of ′cvσ must

fall between ′iσ and ′csσ by proportionality defined by the value of λ (Nelson et al.

CHAPTER 2 54

2012). Nelson et al. (2006) suggested that a reasonable value of λ for the clay soil in the

Front Range area of Colorado, USA, is 0.6. However, the actual value of λ to be used

for a soil should be investigated for that soil.

Nelson and Miller (1992) method suffers from severe shortcomings as it is based on a

depth potential heave (i.e., a single depth below which no heave occurs) and a single

swell pressure value for the complete depth of soil, which is a topic of debate (Singhal

2010).

The oedometer test results are widely used in practice for estimating the heave potential;

however, the environmental factors such as drainage conditions similar to in situ

conditions, and the effects of lateral pressures cannot be simulated well in the oedometer

tests. Attention is also drawn to the possible difficulty in determining a unique swelling

pressure as described by Fredlund et al. (1980) since it is sensitive to the testing

procedure. In addition, this single value of swelling pressure may not be a representative

value over the entire depth of the active zone and for the area considered for expansive

soils heave. The other disadvantage of oedometer methods is the extremely long time

periods required (up to 100 days) for achieving equilibration conditions, which makes

these methods both costly and tedious for use in practice (Holland and Cameron 1981).

2.9.1.2 Empirical methods

To reduce the amount of time required to conduct oedometer tests for estimating the

heave potential, many attempts have been made to correlate oedometer test data with soil

properties. The result of the correlation studies are empirical equations for predicting the

heave potential, accounting both soil state and soil type representative parameters. The

soil state is reflected by placement conditions factors namely moisture content, dry

density, void ratio, and surcharge pressure, while the soil type is reflected by the

compositional parameters namely plasticity index, and clay content (Zumrawi 2013). Rao

et al. (2011) and Vanapalli and Lu (2012) summarized the most common empirical

relations proposed in the literature to correlate the heave potential to the soil properties

(Table 2.4).

LITERATURE REVIEW 55

Table 2.4. The most common empirical methods for the determination of soil heave

potential (modified after Rao et al. 2011 and Vanapalli and Lu 2012) Relationship Reference

5 2.44 3.443.6 10 cS A C−= × Seed et al. (1962)

3 2.442.16 10S PI−= × Seed et al. (1962)

4 2.674.13 10S SI−= × Ranganatham and Satyanarayan (1965)

5 2.67 3.444.57 10 [ / ( 13)]S SI C C−= × − Ranganatham and Satyanarayan (1965)

2 1.452.29 10 ( / ) 6.38p iS PI C w−= × + Nayak and Christensen (1971)

( ) (1 12)(0.4 5.5)nLog S LL w= − + Vijayvergiya and Ghazzally (1973)

( ) (1 19.5)( 0.65 130.5)dLog S LLγ= + − Vijayvergiya and Ghazzally (1973)

0( ) 0.9( ) 1.19pLog S PI w= − Schneider and Poor ( 1974)

0.08380.2558 PIS e= Chen ( 1975)

07.5 0.8 0.203S w C= − + McCormack and Wilding (1975)

0(5.3 (147 ) log ) (0.525 4.1 0.85 )pS e PI P PI w= − − × + − Brackely (1975)

2.77 0.131 0.27 nS LL w= + − O’Neil and Ghazzally (1977)

0 023.82 0.7346 0.1458 1.7 0.00250.00884 , 40

pS PI H w PI wPI H for PI

= + − − + −≥

Johnson (1978)

0 09.18 1.5546 0.08424 0.1 0.04320.01215 , 40

pS PI H w PI wPI H for PI

= − + + + − −≤

Johnson (1978)

4.17 0.386 2.3300.00041( ) ( ) ( )p WS LL P w− −= Weston (1980)

2.559 3.440.0000114 cS A C= Bandyopadhyay (1981)

121.807 (12.1696 ) (27.6579log( ))iS MBV ψ= − + + Cokca (2002)

4.24 0.47 0.14 0.06 55di i iS w q FSIγ= − − − − Rao et al. (2004)

1.1880.6( )PS PI= Azam (2007)

1.71692.0981 ILS e−= Yilmaz (2009)

0log

110

∆

∆ =

+

s

w

fs C w

C

K PHH Ce

Vanapalli et al. (2010)

57.965 37.076 0.524P dS MBVρ ε= − + + + Türköz and Tosun (2011)

1.260.26 0.22 0.7824.5( ) ( ) [ 7.1( ) ( ) ]P iS q PI C F q PI C−= − Zumrawi (2013)

[13,141. where:

CHAPTER 2 56

cA : soil activity C : clay content

sC : swelling index wC : suction modulus ratio e : void ratio 0e : initial void ratio iF : initial state factor

FSI : free swell index H : depth of soil IL : liquidity index K : correction parameter

LL : liquid limit

WLL : weighted liquid limit MBV : methylene blue value P , q : surcharge

fP : final stress state

iq : initial surcharge PI : plasticity index

S , pS , ∆H : heave/swell potential SI : shrinkage index

iw , 0w : initial moisture content

nw : natural moisture content ∆w : change in water content

dρ : in situ dry density

dγ : dry unit weight

diγ : initial dry unit weight

ε : mean-zero Gaussian random error term

iψ : initial soil suction

The comparative studies undertaken by Noble (1966) and Zein (1987) clearly show that

the empirical equations for predicting soil heave potential, while seemingly adequate for

known conditions in the regions where they were developed, have some limitations when

used for other regions. In other words, the empirical relationships are suggested based on

a limited amount of data from a specific region, and it is not appropriate to extend the

usage of these site-specific prediction relationships toward more generalized analysis.

LITERATURE REVIEW 57

Also, these methods evaluate the heave potential for certain defined conditions. For

example, the heave potential of a remolded soil is commonly evaluated for a confining

pressure of 7 kPa and a saturated (zero suction) soil profile (Johnson and Snethen 1978).

Most of the available empirical methods don’t consider some key parameters that

influence the swell behavior such as the soil structure, clay mineralogy, and

environmental factors to list a few. Furthermore, the variation of clay property over the

same site or different sites makes it a challenge to obtain representative soil samples and

determine reliable data from laboratory tests.

Recently, Vanapalli et al. (2010) proposed an empirical method for estimating the

maximum heave potential of natural expansive soils occurring as a response to the water

content variations. The method is based on empirical relationships that have been

developed from the published data of various regions of the world. The information

required for these relationships can be obtained from simple laboratory tests; thus, this

method eliminates the need for difficult and time consuming experimental tests.

Vanapalli et al. (2010) method was tested in seven case studies published in the literature;

however, the method appears to overestimate the field heave for some conditions.

It is suggested that the application of the empirical methods should be used with caution

and should be considered only as indicator for heave.

2.9.1.3 Suction-based methods

Most researchers in the geotechnical engineering field since 1960’s described moisture

movement in unsaturated expansive soils in terms of soil suction (e.g., Richards 1965,

Lytton and Kher 1970, Mitchell 1979, Pufahl and Lytton 1992, Fredlund 1997, Wray

1998, Fredlund and Vu 2001). Richards (1974) suggested that soil suction can be used to

represent the state of the soil water much more effectively than the water content for two

reasons. Firstly, soil suction is primarily controlled by the soil environment and not by

the soil itself, and it tends to not exhibit discontinuous trends. The soil suction profile

tends towards an equilibrium value at a particular depth under a particular climatic

condition while water content is highly sensitive to the soil material variables (e.g., soil

type, clay content, soil density, and soil structure). Secondly, the correlation of soil

CHAPTER 2 58

parameters (i.e., permeability or hydraulic conductivity, diffusivity, and shear strength)

with water content is poor unless other soil properties such as density and clay content

are considered, but these parameters can be conveniently correlated with soil suction.

In suction-based methods, the movement associated with volume change of expansive

soils can be evaluated by measuring the present in situ suction condition and estimating

(or predicting) possible future suction condition under a certain environment. The basic

concept of those methods is that the volume change of unsaturated soils (usually void

ratio or vertical strain) is linearly proportional to the soil suction in a logarithm scale over

the moisture content range between shrinkage limit and plastic limit (Johnson and

Snethen 1978, Mitchell and Avalle 1984, Hamberg 1985). Figure (2.23) shows an

idealized relationship between void ratio and suction for a representative soil sample. The

available suction-based methods differ mainly in the definition of the soil suction

parameter which represents the slope of void ratio versus soil suction plot (e.g., soil

suction index (Johnson 1977, Johnson and Snethen 1978, Snethen 1980, Hamberg 1985,

Dhowian 1990), instability index (Aitchison 1973, Mitchell and Avalle 1984), or suction

compression index (McKeen and Nielsen 1978, McKeen 1980, 1981, 1992, Wray 1984,

1997). The different names for the soil suction parameter arise from the concept that the

unit volume change (i.e., void ratio change) is related linearly to either the soil suction

change or the moisture content change within the range of field conditions. Table (2.5)

lists the most common representatives of suction-based methods.

Fig. 2.23. Idealized void ratio versus logarithm of suction relationship for a representative sample (modified after Hamberg 1985)

LITERATURE REVIEW 59

Table 2.5. Suction-based methods for predicting heave potential (modified after

Vanapalli and Lu 2012) Equation Reference

layers

( - )3 (100 )

f i s

i s

w w GHHw G

∆ = ∑+

where: H∆ : soil heave H : soil layer thickness

iw : initial water content (measured)

fw : final water content (estimated in terms of the equilibrium matric suction)

sG : specific gravity

Richards (1967)

1( )

npt n

iI u zδ

== ∆ ∆∑

where: δ : vertical shrinkage or heave

ptI : instability index ∆u : soil suction change

nz∆ : thickness of the ith soil layer n : total number of soil layers considered

Aitchison (1973)

1

( )n

f i i ii

VS f zV=

∆= ∆∑

( ) log ( ) log ( )f fi h

i i

hVV h σ

σγ γ

σ∆

= − −

where:

fS : surface displacement

if : lateral confinement factor ( )iV V∆ : average volumetric strain

iz∆ : thickness of the ith soil layer n : total number of soil layers considered

ih , fh : initial and final water potentials

fσ : applied octahedral normal stress

iσ : octahedral normal stress above which overburden pressure restricts volumetric expansion

hγ : matric suction compression index

σγ : mean principal stress compression index

Lytton (1977)

CHAPTER 2 60

0

0log

1∆ =

+ +f f

C hH H

e hτ

α σ

( ) (100 )sC G Bτ α=

0 0log ( )h A B w= − where:

∆H : soil heave H : soil layer thickness Cτ : suction index

0e : initial void ratio

0h : matric suction without surcharge pressure

fh : final matric suction

fσ : final applied pressure, (overburden plus external load) α : compressibility index

Johnson and Snethen (1978)

1

[ log( ) log( )]1

ni

t a m a w iii

HH C u C u u

eσ

=

∆ = ∆ − + ∆ −+∑

where:

∆H : soil heave

iH : thickness of the ith soil layer n : total number of soil layers considered

ie : void ratio for the ith layer

tC : compressive index with respect to total stress

mC : compressive index with respect to matric suction ( )auσ − : total stress ( )a wu u− : matric suction

Fredlund (1979)

1( )

npt i

iH I u H

=∆ = ∆∑

ptwyI

w uε∆ ∆

=∆ ∆

where: ∆H : vertical surface movement

ptI : instability index u∆ : soil suction change

iH : soil layer thickness over which ptI can be taken constant n : number of layers to depth of the active zone w∆ : moisture content change

yε∆ : change in vertical strain

Mitchell and Avalle (1984)

LITERATURE REVIEW 61

01

[ log ]1

ni

h ii

HH C h

e=

∆ = × ∆+∑

where:

H∆ : soil heave

iH : thickness of the ith layer

0e : initial void ratio

hC : suction index with respect to void ratio h : soil suction n : number of layers to depth of the active zone

Hamberg (1985)

)(=∆ ∆ − ∆hH H pF pPγ where:

∆H : shrink or swell over vertical increment H : vertical increment over which shrink or swell is occurring

hγ : suction compressibility index pF∆ : change in soil suction over vertical increment pP∆ : change in soil overburden over vertical increment

Wray (1984)

0log

1i

f

CH H

eψ ψ

ψ∆ =

+

( ) (100 )sC G Bψ α=

where: ∆H : soil heave

H : soil layer thickness Cψ : suction index

0e : initial void ratio

iψ , fψ : initial and final suction α : volume compressibility factor

sG : specific gravity of solid particles B : slope of suction versus water content relationship

Dhowian (1990)

hH C u t f s=∆ ∆ ∆ ( 0.02673) ( ) 0.38704hC u w= − ∆ ∆ −

0(1 2 ) 3f K= + 1 0.01(% )s SP= −

where: H∆ : surface heave

hC : suction compression index u∆ : suction change t∆ : soil layer thickness

McKeen (1992)

CHAPTER 2 62

f : lateral restraint factor s : reduction factor to account for overburden

w∆ : moisture content change

0K : coefficient of lateral earth pressure at rest SP : swell pressure applied to the soil due to overburden

pressure

1

( )n

f i i ii

VS f zV=

∆= ∆∑

( ) log ( ) log ( )∆= − −f f

i hi i

hVV h σ

σγ γ

σ

( ) ( )( )2h

swelling case shrinkagecaseγ γγ +=

3( ) [( 1) 1]100

COLEswelling caseγ = + −

3

1( ) [1 ]( 1)

100

shrinkagecaseCOLE

γ = −+

where: fS : surface displacement

if : lateral confinement factor ( )iV V∆ : average volumetric strain

iz∆ : thickness of the ith soil layer n : total number of soil layers considered

ih , fh : initial and final water potentials

fσ : applied octahedral normal stress

iσ : octahedral normal stress above which overburden pressure restricts volumetric expansion

hγ : matric suction compression index

σγ : mean principal stress compression index COLE : coefficient of linear extensibility

Cover and Lytton (2001)

1

( )n

i ii

VH f zV=

∆∆ = ∆∑

,( ) log ( ) log ( )f fi swelling h

i i

hVV h σ

σγ γ

σ∆

= − −

,( ) log ( ) log ( )f fi shrinkage h

i i

hVV h σ

σγ γ

σ∆

= − +

0.67 0.33f pF= − ∆ 0/ (1 )cC eσγ = +

where: H∆ : surface displacement

Lytton et al. (2004)

LITERATURE REVIEW 63

f : crack fabric factor, 1/ 3 1.0f≤ ≤ ( )iV V∆ : volume strain

iZ∆ : the ith depth increment n : number of depth increments

ih , fh : initial and final values of matric suction

iσ , fσ : initial and final values of mean principal stress

hγ : matric suction compression index

σγ : mean principal stress compression index pF∆ : change of suction

cC : compression index

0e : void ratio

Because of greater sensitivity of soil suction to volume change in comparison to moisture

content, soil suction-based methods have provided better characterization of expansive

soil behavior and more reliable estimates of anticipated heave under field conditions than

oedometer methods (Johnson and Snethen 1978, Snethen 1980, Fredlund 1983). The

heave calculation procedure of suction-based methods requires both the initial and final

profiles of suction along with the suction parameter. The values of initial and final

suction are either estimated during some correlation (e.g., equilibrium soil suction and

climate index correlation) or measured using various suction devices (e.g., tensiometers,

thermocouple psychrometer, thermal conductivity sensors, filter paper, etc.) (Snethen

1980). Fredlund (1979) suggested that the initial suction is measured by using one of

suitable devices; however, assumptions can be used for the final suction profile. By

assuming the pore air pressure is initially to be in equilibrium and equal to atmospheric

pressure, the pore water pressure can be equivalent to the matric suction, and then three

different scenarios can be used for determining the final pore water pressure (i.e., the

final matric suction): (i) assuming the groundwater table at the soil surface, thus a

hydrostatic pore water pressure will result; (ii) the pore water pressure approaches zero

throughout its depth; and (iii) the pore water pressure is slightly negative. These

assumptions are simple and reasonable, but they do not provide details of how soil

volume changes with respect to time.

Suction-based methods are generally considered to be simple, economical, expedient and

capable of simulating field conditions. However, several possible limitations summarized

below are likely when they are extended in practice:

CHAPTER 2 64

- The relationship of the soil suction on log scale versus volume change is linear only

over a certain range of suction; this range is not known or well defined in most cases.

- Soil suction has to be measured or estimated for each site. However, it is a challenge

to reliably measure suction in the field especially in expansive soils.

- Although the parameters of suction-based methods (e.g., suction compression index)

are functions of stress states (i.e., net normal stress and matric suction), the

determination of these parameters has been done by controlling both the net normal

stress and the matric suction using simple laboratory index tests.

- The prediction accuracy of suction-based methods depends on the determination of

wetting depth (i.e., depth of suction variations). However, the depth of wetting is

difficult to define as it varies with site and depends on the environmental changes

(Snethen and Huang 1992).

2.9.2 Methods for predicting soil vertical movement over time

The soil movement over time information is required for the design of foundations placed

in expansive soils. This information is also helpful for the assessment of pre-wetting and

controlled wetting mitigation alternatives for expansive soils. Research particularly in the

past decade has been directed by various investigators to propose methods for the

prediction of the soil movement over time (e.g., Briaud et al. 2003, Vu and Fredlund

2004 and 2006, Zhang 2004, Wray et al. 2005, Overton et al. 2006, Nelson et al. 2007).

Briaud et al. (2003) suggested that any method developed to predict the movement of

expansive soils over time must include two components: (i) the range of water content or

soil suction fluctuations as a function of time within the active zone depth; and (ii) the

constitutive law that links soil state variable (i.e., water content, soil suction, and

mechanical stress) to the volume change movements of the soil. The current methods of

the prediction of soil movement over time can be classified into: (i) consolidation theory-

based methods that use the matric suction and the mechanical stress as state variables

(i.e., extending two independent stress state variables concept proposed by Fredlund and

Morgenstern (1977)), (ii) water content-based methods that use water content as a state

variable, and (iii) suction-based methods that use the matric suction as a state variable.

The following describes some of these methods.

LITERATURE REVIEW 65

2.9.2.1 Consolidation theory-based methods

Vu and Fredlund (2004) method

Vu and Fredlund (2004) extended the general consolidation theory of unsaturated soils

described in section (2.8.3) to develop a method for the prediction of one-, two-, or three-

dimensional soil heave over time. The governing equations for soil structure (i.e.,

equilibrium equation, Equation 2.25) and for water phase (i.e., water continuity equation,

Equation 2.30) were numerically solved using uncoupled and coupled analyses. In the

uncoupled analysis, equation (2.25) was solved independently from equation (2.30). A

general-purpose partial differential equation solver (FlexPDE) was used to obtain the

uncoupled solution. Two key steps are used in FlexPDE; (i) soil matric suction

distribution with respect to time for specified boundary conditions is determined; (ii) the

soil heave is then determined taking account of the applied boundary conditions and the

matric suction variations. However, in the coupled analysis, the governing equations were

solved simultaneously using finite element computer program COUPSO (Pereira 1996).

The results include the soil heaves and the matric suctions obtained at any time during the

transient process. The uncoupled solutions can be more easily achieved than the coupled

solutions because of the nonlinear functions of soil properties involved in each process

(i.e., water flow or stress deformation) are considered to be independent of one another.

A case history of a floor slab of a light industrial building located in Regina,

Saskatchewan, Canada, was modeled by Vu and Fredlund (2004) to test the validity of

their prediction method. Figure (2.24) provides the comparison of the soil heave

predicted by Vu and Fredlund (2004) at various suction conditions over time with the

total heave measured by Yoshida et al. (1983) at different depths beneath the centre of the

slab. The agreement between the predicted and the measured heaves differs to some

degree. The amounts of heave measured at depths of 0.58 m and 0.85 m correspond to the

predicted heave at 100 days. Figure (2.25) shows the comparison between the predicted

heave at various matric suction conditions over time and the measured total heave at the

surface of the slab. The total heave predicted under the steady state condition agrees well

with the measured heave. The predicted results are in a reasonable agreement with the

measured values.

CHAPTER 2 66

Fig. 2.24. Measured and predicted heaves with depth under the center of the slab (modified after Vu and Fredlund 2004)

Fig. 2.25. Measured and predicted heaves at the surface of the slab (modified after Vu and Fredlund 2004)

Vu and Fredlund (2006) investigated challenges encountered by Vu and Fredlund (2004)

to characterize the void ratio at low net normal stresses and (or) low matric suctions.

Extremely low elastic moduli are possible for low net normal stresses or low matric

suctions values which contribute to unreasonably large soil movements. These challenges

LITERATURE REVIEW 67

have been overcome by providing a continuous, smooth void ratio constitutive surface

based on the soil swelling indices obtained from conventional oedometer tests. Two

typical volume change problems, water leakage from a pipe under a flexible cover and

water infiltration at the ground surface, were solved by Vu and Fredlund (2006) using

both the coupled and uncoupled analyses. It was suggested that the uncoupled analysis

may be adequate for most heave prediction problems. However, the coupled analysis

provides a more rigorous understanding of the swelling behavior of expansive soils and

forms a reference for the evaluation of various uncoupled analyses.

The prediction method presented in Vu and Fredlund (2004, 2006) has been validated

using Regina expansive clay. The method focuses on the prediction of soil heave, which

is corresponding to a short-term condition. The soil shrinkage corresponding to a long-

term condition is completely neglected. In addition, the coefficients of volume change

1sm , 2

sm , 1wm , and 2

wm , which are needed to perform the uncoupled/coupled analyses,

require the void ratio and water content constitutive surfaces to be constructed. Those

constitutive surfaces can be obtained from the consolidation tests or the triaxial shear

tests with suction control. However, such tests are usually time consuming and require

advanced lab equipments.

Zhang (2004) method

Unsaturated soils attain the saturated condition under different scenarios; however,

researchers were unable to provide a unified theoretical framework for both saturated and

unsaturated soils. Several investigators provided the coupled consolidation theory for

saturated and unsaturated soils, separately. The concept of constitutive surfaces however

has been provided only for unsaturated soils because of the following reasons (Zhang et

al. 2005): (i) the volume change theory for saturated soils is well established through the

research work of Terzaghi (1936) and Biot (1941). Both Terzaghi (1936) and Biot (1941)

suggested using the consolidation curve and did not use the constitutive surfaces for

saturated soils; (ii) many researchers have used a single stress state variable (i.e., the

effective stress principle) for interpreting the behavior of saturated soils.

CHAPTER 2 68

Zhang (2004) provided the coupled consolidation for saturated and unsaturated soils in a

unified manner. The thermodynamic analogue was used to explain the coupled

consolidation process for saturated and unsaturated soils following Terzaghi (1943)’s one

dimensional consolidation theory for saturated soils. Terzaghi (1943) stated that if the

unit weight of water wγ is assumed to be unity, the differential equation of Terzaghi’s

consolidation theory is identical to the differential equation for non-stationary, one-

dimensional flow of heat through isotropic bodies. The loss of water (consolidation)

corresponds to the loss of heat (cooling) and the absorption of water (swelling) to the

increase of heat content of a solid body (Zhang 2004). In other words, the pore water

pressure corresponds to the temperature while the water content to the heat energy per

unit mass. The coupled consolidation theory for saturated and unsaturated soils includes

the differential equations for soil structure and water phase. The differential equation for

soil structure is given in equation (2.25). However, to derive the differential equation for

water phase, it has been assumed that the continuity equation for water phase is similar to

that for heat transfer (i.e., using the thermodynamics principles). As a consequence, the

differential equation for water phase can be written in terms of specific water capacity of

a soil (i.e., the volume of water required decreasing unit mass of soil by 1 kPa of matric

suction).

2( ) ( )w x ymean a a w w w

d w w ww w

d u d u u u uC m k Y k Yt t x x g y y g

σρρ ρ

− − ∂ ∂ ∂ ∂+ = + + + ∂ ∂ ∂ ∂ ∂ ∂

z ww

w

uk Yz z gρ

∂ ∂+ + ∂ ∂

(2.36)

where dρ is dry density of a soil, and wC is specific water capacity of a soil.

Equations (2.25) and (2.36) are the differential equations for the coupled hydro-

mechanical stress (consolidation) problem for unsaturated soils. However, by using the

constitutive surfaces proposed by Zhang (2004) for saturated and unsaturated soils,

equations (2.25) and (2.36) can also be used for saturated soils as a special case. Two

stress state variables (i.e., total stress and pore water pressure) were used for saturated

LITERATURE REVIEW 69

soils in order to develop the constitutive surfaces for both saturated and unsaturated soils

mechanics in a unified system with smooth transition.

Close examination shows equations (2.30) and (2.36) are the same; the left sides of both

equations are the volumetric water content variation and the right sides represent the net

water flow into the soil element. If equation (2.36) is used, the water generation could be

easily simulated by the heat generation based on the thermodynamic analogue.

Consequently, some already well-established commercial software packages (e.g.,

ABAQUS, SUPER, and ANSYS) for solving those coupled thermal stress problems

could be modified for solving the coupled consolidation problem to simulate several

complicated problems related to the geotechnical engineering.

Zhang (2004) modelled a site in Arlington, Texas, USA, extending the coupled

consolidation theory of saturated-unsaturated soils for the estimation of the soil

movement over time. Four full-scale spread footings, called RF1, RF2, W1, and W2,

constructed on expansive soils of the Arlington site were modelled over a period of 2

years. Factors that influence the movements of expansive soils such as the daily weather

data, including the daily temperature, solar radiation, relative humidity, wind speed and

rainfall, and the vegetation were considered for this field construction site.

Abaqus/Standard program was used for the simulation of the soil movements at the

Arlington site based on several models, including the coupled consolidation theory for

saturated-unsaturated soils, potential and actual evapotranspiration estimation by using

daily weather data, theories for the simulation of the soil-structure interaction at the soil-

slab interface. The same site was also modeled by Briaud et al. (2003) to investigate the

damage caused by expansive soils with both concrete and asphalt pavements. More

details of the Arlington site are available in Briaud et al. (2003) and Zhang (2004).

For the simulation of soil volume change behavior, a coupled hydro-mechanical stress

analysis was used and thermodynamic part was corresponded to the water phase

continuity of the soil. By applying initial and boundary conditions and using the finite

element method to solve the differential equations of the coupled consolidation theory

(Equations 2.25 and 2.36), the values of mechanical stress and the matric suction can be

CHAPTER 2 70

calculated. The estimated values of mechanical stress and matric suction were used as an

initial condition for the next step. This simulation technique can be continuously

performed to predict the soil movement over time.

The validation of the prediction method was assessed more closely by comparing its

estimations of soil movement with the long term field observations (over 2 years) at the

Arlington site. Figure (2.26) shows the average values of the predicted soil movements at

the four corners of the modeled footing, the measured movements of the four footings,

and the average values of the measured movements of the footings over the two year’s

period. The comparison of the predicted movements with the measured movements of

each footing did not lead to as good as the comparison based on the average values of the

measured movements of the four footings (see Figure 2.26). This could be attributed to

the fact that the regular measurements of soil movement can vary significantly while the

average measurements are more representative of the actual variation in the movement of

the soil site (Zhang 2004). The results suggested that Zhang (2004) method based on the

coupled consolidation theory of saturated-unsaturated soils is a valuable tool for

predicting the soil movements over time.

Fig. 2.26. Soil movement predicted by Zhang (2004) method and the soil movements measured at the Arlington site over two years (modified after Zhang 2004)

LITERATURE REVIEW 71

Zhang (2004) method is a comprehensive approach for modeling the water flow and the

soil movement over time. Complex numerical solutions in finite element computer

programs are required in this approach to address the analogy between the thermal and

hydraulic problems. The use of the constitutive surfaces in this approach contributed to

using a unified system for the first time to simulate the volume change behavior of

expansive soils under both saturated and unsaturated conditions. However, there are

limitations to apply the three-dimensional constitutive surface model in practice. The

proposed constitutive surfaces have been developed based on testing soils under

conditions not typically experienced in the field such as a shrinkage or a matric suction

test at no normal stress, or a consolidation test at fully saturated conditions. Also, the

conventional laboratories are not equipped to conduct the shrinkage tests or the matric

suction tests which are needed for constructing the constitutive surfaces of unsaturated

soils. These tests are time consuming that require sophisticated laboratory equipment and

trained personnel, and hence such an approach is expensive.

2.9.2.2 Water content-based methods

Briaud et al. (2003) method

Briaud et al. (2003) proposed a method for estimating the vertical movement

(shrink/swell) of the ground surface due to the variations in soil water content over time.

The soil water content was used as a governing parameter; the range and the depth of

water content variations can be estimated from a combination of experience, databases,

observations, and calculations. The shrink test was suggested to obtain the relationship

between the change in water content and the volumetric strain induced. Figure (2.27)

shows the typical relationship of the water content versus the volumetric strain obtained

from the shrink test. This relationship can be approximated by a straight line with the

slope being the shrink–swell modulus Ew. The procedure of the method can be described

as follows:

- Determine the depth of water content fluctuation and break it into an appropriate

number of n layers, Hi being the thickness of the layer i.

- Collect soil specimens at the site within the depth of water content fluctuation.

CHAPTER 2 72

- Perform the shrink test on each of the specimens; determine the shrink-swell modulus

wE and the shrinkage ratio f (i.e., the ratio of the vertical strain to the volumetric

strain.)

- Determine the change in water content w∆ as a function of depth and time.

- For the ith layer, calculate the vertical movement of that soil layer iH∆ as

/i i i i wiH H f w E∆ = ∆ (2.37)

- Add the vertical movements of all soil layers for each date to calculate the ground

surface movement for a given time as

1 1 1( / )

n ni i i i wii

H H H f w E= =∑ ∑∆ = ∆ = ∆ (2.38)

Fig. 2.27. Soil water content versus volumetric strain obtained from the shrink test (modified after Briaud et al. 2003)

This shrink test-water content method was evaluated by comparing the predictions with

the measurements of the soil movements at the four full-scale spread footings (i.e., RF1,

RF2, W1, and W2) constructed in the Arlington site; the same site that was modeled by

Zhang (2004) (see section 2.9.2.1). Specimens were taken at the site during the 2-year

period and data including water content and shrink-swell modulus were measured. Figure

(2.28) shows the comparison between the soil movements predicted by Briaud et al.

LITERATURE REVIEW 73

(2003) method and the measured movements of footings over two years. A better

simulation of the field soil movements was achieved using Zhang (2004) method in

comparison to Briaud et al (2003) method.

Fig. 2.28. Soil movements predicted by Briaud et al. (2003) method and the measured soil movements at the Arlington site over two years (modified after Briaud et al. 2003) The advantage of Briaud et al. (2003) method however is its capability to predict the

vertical soil swelling and soil shrinkage simultaneously using the shrink test. This method

is based on the information of water content which is more reliable and simpler to

measure in comparison to the soil suction. The constitutive law is obtained from the

shrink test conducted on site-specific specimens instead of correlations to the index

properties. However, the method is an uncoupled analysis of unsaturated soils where only

the influence of moisture variation on the volume change of expansive soils is

considered. In addition, when the soil is highly fractured, the shrink test is difficult to

perform. Another drawback is that any theoretical consideration must make the use of the

soil-water characteristic curve to transform the governing equations from suction based

equations to water content based equations (Briaud et al. 2003).

Overton et al. (2006) method

The amount of soil heave at any time depends on two factors. These are the depth at

which the water content in the soil has increased over time, and the expansion potential of

CHAPTER 2 74

the various soil strata. As water migrates through a soil profile, different strata become

wet, some of which may have more swell potential than others. Consequently, the amount

of soil heave varies with time. Overton et al. (2006) presented an approach for predicting

the free field heave of expansive soils over time based on the migration of the wetting

front. Analyses of the migration of the wetting front were conducted for soil profiles

using the commercial software VADOSE/W (Geo-Slope 2005). VADOSE/W is a finite

element program that can be used to model both saturated and unsaturated flow in

response to changes in the atmospheric conditions while considering infiltration,

precipitation, surface water runoff and ponding, plant transpiration and actual

evaporation, and heat flow. The free field heave, which will occur at the ground surface if

no stress is applied, is the fundamental parameter required in this approach. The free-field

heave was predicted using the oedometer method of Nelson and Miller (1992) presented

in section (2.9.1.1).

By assuming various values of swelling pressure and percent swell, the maximum free

field heave and the depth of heave potential can be calculated using equation (2.34). The

amount of the heave at any point in the soil profile is a function of the amount by which

the water content has increased. The values of the volumetric water content are obtained

from VADOSE/W at each time step. The relationship between the heave potential and the

volumetric water content for a soil can be determined from oedometer tests conducted in

the laboratory. For soils that are not fully wetted, the percent swell and the swelling

pressure will be less than those measured after saturation in the oedometer tests.

Therefore, in calculating the soil heave, those values must be corrected for the actual

volumetric water content. The free field heave with respect to time is computed by

multiplying the total heave potential (i.e., maximum free heave from equation (2.34)) at

each soil layer by a heave factor obtained from the heave potential and volumetric water

content relationship.

Overton et al. (2006) extended this approach on soil profiles in the Denver area of

Colorado with good and poor drainage. Chao et al. (2006) also used this approach to

investigate the effect of irrigation practices, poor drainage conditions, deep wetting from

underground sources, and dipping bedrock on the heave variations over time. The results

LITERATURE REVIEW 75

showed significant variations exist in the predicted values of heave potential versus time

due to the effect of those factors.

Realistic estimates of the time rate of the migration of the wetting front and the resulting

soil heave can be obtained by Overton et al. (2006) method for only ideal conditions.

Such ideal conditions are only possible where sites have homogenous soil profiles with

minimal macroscale fracturing or cracking, and/or where the principal direction of heave

is perpendicular to the ground surface. However, if the site specific analyses have not

accurately determined the rate of migration of the wetting front and the resulting time rate

of heave, the entire depth of heave potential should be assumed wet during the life of a

structure (i.e., maximum heave potential should be considered) (Overton et al. 2006). In

addition, experimental determination of the free-field heave using oedometer tests is both

time consuming and difficult to conduct. Some downsides to oedometer tests are related

to the extremely long time period required for achieving the equilibrium condition and

the difficulty to simulate the in situ conditions (e.g., drainage conditions and lateral

pressures). Another drawback of this method is that it doesn’t give any indication of

possible shrinkage.

2.9.2.3 Suction-based methods

Wray et al. (2005) method

Wray et al. (2005) developed a computer program SUCH (it is named from SUCtion

Heave) to predict the soil moisture changes and the resulting soil surface movements

(heave/shrink), particularly under covered surfaces. The SUCH program involves two

models: (i) a moisture flow model for estimating the movement of water through

unsaturated expansive soils based on the diffusion equation developed by Mitchell

(1993), and (ii) a volume change model developed by Wray (1997) for estimating the

vertical soil movement (heave/shrink) associated with the change in soil suction over

time.

The Mitchell’s transient suction diffusion equation in its three dimensional takes the form

2 2 2

2 2 2

( , , , ) 1u u u f x y z t ux y z p tα

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂ (2.39)

CHAPTER 2 76

where u is total soil suction expressed in pF units (kPa = 0.1 × 10pF), α is diffusion

coefficient (mm2/s) which can be measured in the laboratory (Mitchell 1993) or

calculated from empirical equations (McKeen and Johnson 1990, Bratton 1991, Lytton

1994), p is unsaturated permeability (mm/s), t is time (s), x, y, and z are space

coordinates, and f(x, y, z, t) is internal source of moisture.

SUCH program is written in FORTRAN language, utilizing the finite difference

technique to solve the transient suction diffusion equation (Equation 2.39). Two sets of

information must be given: (i) the initial condition, i.e. the initial value of suction at each

node in the soil mass; and (ii) the boundary conditions, i.e. the values of suction on the

boundaries of the soil mass at each time step. Then, the moisture flow model can be used

to determine the distribution of soil suction in the soil mass over time.

After the determination of the suction distribution through the unsaturated expansive soil

mass, the resulting vertical soil movement at each nodal point associated with the change

of soil suction over time can be estimated. The suction-based model (Equation 2.40)

developed by Wray (1997) was used for the estimation of the resulting soil movements.

, ,, , , , , ,[ ]∆ = ∆ ∆ − ∆i j ki j k h i j k i j kH z pF pPγ (2.40)

where , ,i j kH∆ is incremental volume change (heave/shrink) at grid point (i, j, k) over the

increment thickness z∆ , z∆ is increment thickness in the z-direction over which heave or

shrink occurs, , ,i j khγ is suction compression index at grid point (i, j, k), McKeen (1980),

Lytton (1994), and Wray (1997) presented different methods to estimate the value of

, ,i j khγ , , ,i j kpF∆ is change of total soil suction expressed in pF units at grid point (i, j, k),

and , ,i j kpP∆ is change of soil overburden over the increment thickness z∆ at grid point (i,

j, k). The vertical movement of each nodal point at the top surface of the soil mass was

calculated as the summation of the vertical movements of the nodal points on the vertical

line passing through that surface point, extending from the top to the bottom of the active

zone of the soil mass (Wray et al. 2005).

LITERATURE REVIEW 77

The method was validated using well-documented field studies, chosen to cover widely

varying climatic and soil conditions, that are located in the United States and Saudi

Arabia. Two sites; namely, Amarillo test site and College Station test site, located in

Texas, USA, were selected to represent a three dimensional problem (Wray 1989). The

College Station site and the Amarillo site properties are similar. The only exception is

that the College Station site represents a wet climate while the Amarillo site was selected

to represent a dry climate. SUCH model was used for the two sites to estimate the soil

suction changes and the vertical soil movements every month over a period of 5 years

(from August 1985 through July 1990). Al-Ghatt, Saudi Arabia, test site investigated by

Dhowian et al. (1985) was also selected to represent a two-dimensional problem. The test

site was modeled over a period of 36 weeks. Comparisons were made between the

estimated and the measured soil surface movements at several locations for the field

studies under consideration. Figure (2.29) shows the predicted and measured monthly

surface movements at 1.8 m outside slab edge along longitudinal axis at Amarillo site.

Fig. 2.29. Predicted and measured monthly surface movements at 1.8 m outside slab edge along the longitudinal axis at Amarillo site (modified after Wray et al. 2005)

The results of the SUCH model have shown a moderate to a good correlation with the

reported field measurements of soil suction and the associated soil movements for the

three sites (Wray et al. 2005). However, the application of the SUCH model to practical

CHAPTER 2 78

problems depends on the quantitative expression of the model parameters (i.e., diffusion

coefficient, equilibrium soil suction, active zone depth, and suction compression index)

and the initial and boundary conditions. Consequently, if the model parameters and the

initial and boundary conditions can be accurately determined, the soil suction distribution

and the resulting soil surface movements can be reasonably reproduced by the computer

model SUCH. In addition, the results of the validation process revealed that Mitchell’s

diffusion equation for soil suction (Equation 2.39) needs to be modified to model the

moisture movements in unsaturated fissured soils (soil cracks mechanism upon wetting).

In SUCH model, the initial soil suction value information is required for each site.

However, it is a challenge to reliably measure the field suctions especially in expansive

soils.

2.10 Summary

Significant advances have been made towards the prediction of the heave and the shrink

related volume change behavior of expansive soils since 1960’s. The focus of most

prediction methods has been towards estimating the maximum heave potential (soils

under the saturation condition). These methods are classified into: empirical methods,

oedometer methods, and suction-based methods. These methods are valuable; however,

they do not provide information about the field soil movements as a function of time.

Research studies show that the predicted soil heave by using the assumption of the

saturation condition as a limiting condition is much higher than the in situ expansive soil

heave. The soil moisture changes due to the environmental variations or other factors

have a significant influence on the soil movement over time. Due to this reason,

information related to the soil movement with respect to time is of practical interest for

both the reliable design of foundations for structures on expansive soils and the

assessment of mitigation alternatives for expansive soils.

In an attempt to develop both reliable and economical procedures of soil volume change

analysis, few studies in recent years proposed prediction methods for estimating the

expansive soil movements over time. Table (2.6) summarizes those recent methods which

are classified into three main categories: consolidation theory-based methods, water

LITERATURE REVIEW 79

content-based methods, and suction-based methods. However, there are limitations to

apply the available methods in practice. The current volume change constitutive

relationships are developed based on testing the soils under conditions not experienced in

the field. Also, many conventional laboratories are not equipped to run the suction-

controlled tests that are required for determining the parameters of the constitutive

relationships. These tests generally require costly, time consuming, and difficult

laboratory testing. Most importantly, few prediction methods have been validated with

field measurements using one or limited number of case studies.

The prediction of volume change movements is a challenge even to date to geotechnical

engineers. The available literature reviewed in this chapter highlights the need for a

simple and efficient method that can be easily used in the conventional engineering

practice to reliably predict the expansive soil movements with respect to environmental

changes over time.

CHAPTER 2 80

Table 2.6. Summary of the current methods for predicting the volume change movement of expansive soils over time

Description

Consolidation theory-based methods Water content-based methods Suction-based methods Vu and Fredlund (2004) Zhang (2004) Briaud et al. (2003) Overton et al. (2006) Wray et al. (2005)

Governing equations

Water continuity eq. (Eq. 2.30) Stress equilibrium eq. (Eq. 2.25)

Water continuity eq. in terms of

wC (Eq.2.36) (using thermodynamic analogue) Stress equilibrium eq. (Eq.2.25)

The constitutive equation of the soil movement formulated by extending the parallel between the shrink test–water content method and settlement methods (Eq. 2.38)

Free field heave eq. (Eq. 2.34) Mitchell’s transient suction diffusion eq. (Eq. 2.39) Suction-based model (Eq. 2.40)

State variables Matric suction, a w(u u )− and net normal stress, mean a( u )σ −

For saturated soils: total stress, σ and pore water pressure, wu For unsaturated soils: a w(u u )− ,

mean a( u )σ −

Gravimetric water content, w Volumetric water content, wθ Total soil suction

Required tests Conventional oedometer test, oedometer or triaixal shear tests with suction control

Consolidation-swell test, free shrink test, suction test, and specific gravity test

Shrink test Filter paper tests, consolidation-swell test, constant-volume test

Tests to measure the diffusion coefficient and the suction compression index

Soil properties µ = Poisson’s ratio Elasticity moduli:

a w mean aE fn[(u u ), ( u )]= − −σ

a w mean aH fn[(u u ), ( u )]σ= − −

w a w mean aE fn[(u u ), ( u )]σ= − −

w a w mean aH fn[(u u ), ( u )]σ= − −Permeability function:

w a w mean ak fn[(u u ), ( u )]σ= − −

µ E = saturated modulus of elasticity

dρ = soil dry density α = coefficient of expansion

wC = specific water capacity w2m = coefficient of water

volume change with respect to

a w(u u )− Permeability function

w a w mean ak fn[(u u ), ( u )]σ= − −

wE = shrink-swell modulus f = shrinkage ratio

HC = heave index SWCC = soil water characteristic curve

satk = saturated hydraulic Conductivity, or hydraulic conductivity function :

w a wk fn[(u u )]= −

α = diffusion coefficient p = unsaturated permeability γ = suction compression index

Computer program

FlexPDE for uncoupled analysis COUPSO for coupled analysis

ABAQUS with three models: (i) coupled consolidation theory for saturated-unsaturated soils, (ii) potential and actual evapotranspiration estimation, and (iii) theories for the simulation of the soil-structure interaction

None VADOSE/W for simulating water migration in response to atmospheric conditions

SUCH with two models: (i) moisture flow model, and (ii) a volume change model

LITERATURE REVIEW 81

Initial condition Initial matric suction, a w i(u u )− and initial net normal stress,

mean a i( u )σ −

a w i(u u )− , mean a i( u )σ − None Initial water content profile Initial total suction

Boundary conditions

Matric suction, water flux, applied load, and soil displacements

Matric suction, water flux, applied load, soil displacements, climate data, and vegetation data

None Water flux, pore water pressure, climate data

Total suction

Results 1-D, 2-D, and 3-D heave over time

Vertical movement of the ground surface (heave/ shrink) over time

Vertical movement of the ground surface (heave/shrink) over time

free field heave over time soil surface movements (heave/shrink) under covered surfaces

Application A light industrial building in Regina, Saskatchewan, Canada

To simulate footings’ movements for a site in Arlington, Texas, USA

To simulate footings’ movements for a site in Arlington, Texas, USA

Soil profiles in the Denver area of Colorado, USA

Amarillo test site and College Station test site located in Texas, USA, and Al-Ghatt, Saudi Arabia, test site

CHAPTER 2 82

CHAPTER 3

MODULUS OF ELASTICITY OF UNSATURATED

EXPANSIVE SOILS

3.1 Introduction

The modulus of elasticity is conventionally used in geotechnical engineering practice

in the calculation of the soil stress and deformation behavior (Davis and Poulos 1968,

Schmertmann 1970, Schmertmann et al. 1978, Bowles 1987, Lade and Nelson 1987,

Lade 1988, Berardi and Lancellotta 1991, Lancellotta 1995, Terzaghi et al. 1996, Mayne

and Poulos 1999, Lee et al. 2008, Akbas and Kulhawy 2009, Oh et al. 2009, Vanapalli

and Mohamed 2013). Prior to all these studies, Terzaghi’s (1925, 1926, and 1931)

pioneering modeling and experimental studies showed that the swelling and shrinkage of

clay soils are essentially elastic deformations. It was shown that the swelling capacity of

any soil is dependent on the elastic properties of the solid phase of the soil. Similar

conclusions were derived by several investigators using different approaches (Biot 1941,

Coleman 1962, Matyas and Radhakrisha 1968, Barden et al. 1969, Aitchison and

Woodburn 1969, Brackley 1971, Aitchison and Martin 1973, Fredlund and Morgenstern

1976 and 1977, Vu and Fredlund 2004 and 2006, Zhang 2004, Zhang and Briaud 2010).

The total or effective stress, on the other hand, can modify the pore and soil skeleton

structures and have a significant influence on the soil stiffness (the soil modulus of

elasticity). However, under unsaturated conditions, the effective stress or the stress in soil

skeleton is affected not only by the total stress, but also by the inter-particle stresses. The

inter-particle stresses in an unsaturated soil are also referred to as suction stresses that

arise due to the variation in soil water content or matric suction. The suction stresses

consist of inter-particle physicochemical forces and pore water attraction due to matric

suction (Lu and Likos 2006). Therefore, in fine-grained soils (e.g., silty and clayey soils),

83

the modulus of elasticity is significantly influenced by soil water content or matric

suction (Fredlund and Rahardjo 1993, Costa et al. 2003, Inci et al. 2003, Lu and Likos

2006,Yang et al. 2008).

Vanapalli and Oh (2010) proposed a semi-empirical model for predicting the modulus of

elasticity of unsaturated soils with respect to matric suction using the soil-water

characteristic curve (SWCC) as a tool. The validity of the model was tested for coarse-

and fine-grained soils with plasticity index Ip values lower than 16%. For expansive soils

with higher plasticity index, the adaptability of the Vanapalli and Oh (2010) has not been

reported in the literature yet.

In this chapter, the validity of Vanapalli and Oh (2010) model to predict the modulus of

elasticity of unsaturated expansive soils has been assessed using triaxial shear test results

from the literature.

3.2 Background

Oh et al. (2009) proposed a semi-empirical model for predicting the variation of modulus

of elasticity of unsaturated coarse-grained soils using the SWCC and the modulus of

elasticity under saturated condition, extending similar concepts that were followed for the

prediction of shear strength (Vanapalli et al. 1996) and bearing capacity (Vanapalli and

Mohamed 2007) of unsaturated soils. The model was developed by Oh et al. (2009) using

the stress versus displacement relationships from model footing tests performed on

different sands under unsaturated conditions. In this model, as shown in equation (3.1),

two fitting parameters α and β were used.

( )1/101.3a w

unsat sata

u uE E S

Pβα

− = +

(3.1)

where Eunsat is soil modulus of elasticity under unsaturated condition, Esat is soil modulus

of elasticity under the saturated condition, ( )a wu u− is matric suction, Pa is atmospheric

pressure (Pa = 101.3 kPa) used for maintaining consistency with respect to the

dimensions and units on both sides of the equation, and S is degree of saturation. Based

CHAPTER 3 84

on the results of Oh et al. (2009), the fitting parameter β = 1 was required for coarse-

grained soils (i.e., plasticity index Ip = 0%). The fitting parameter α was found to be a

function of footing size; values between 1.5 and 2 were recommended for large size

footings to reasonably estimate the modulus of elasticity of unsaturated sandy soils Eunsat.

The differential form of equation (3.1), shown in equation (3.2), was used for explaining

the nonlinear variations of modulus of elasticity with respect to matric suction.

( ) ( ) ( )( )

( )( /101.3)

ββα

= + −− −

unsat sata w

a w a a w

d SdE E S u ud u u P d u u

(3.2)

Equation (3.2) shows, in coarse-grained soils, the net contribution of matric suction

towards the increase in the modulus of elasticity starts decreasing as the matric suction

approaches the residual suction value. Such a behavior can be attributed to both the low

value of the degree of saturation S and the negative value of ( ) ( )a wd S d u uβ − at the

residual condition (Figure 3.1) (Oh et al. 2009).

Fig. 3.1. The relationship between (a) soil-water characteristic curve (SWCC), (b) the variation of modulus of elasticity with respect to matric suction (modified after Oh et al. 2009)

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 85

Review of Figure (3.1) shows that the modulus of elasticity linearly increases up to the

air-entry value of the soil; beyond this value and up to the residual suction value there is a

nonlinear increase in the value of the modulus of elasticity. However, the contribution of

matric suction towards the modulus of elasticity starts to decrease beyond the residual

suction value for coarse-grained soils (Oh et al. 2009). This can be attributed to the

variation of area of water with desaturation over the different stages of the soil-water

characteristic curve (SWCC) (Vanapalli 1994, Vanapalli et al. 1996). Figure (3.1a) shows

the three identifiable stages of desaturation, namely: the boundary effect stage, the

transition stage, and the residual stage. In the boundary effect stage, all the soil pores are

filled with water. The soil is essentially saturated, and there is no reduction in the area of

water. In this stage, the single stress state ( )wuσ − describes the behavior of the soil.

Hence, there is a linear increase in the modulus of elasticity up to the air -entry value at

which air starts to enter the largest pores of the soil. Then, in the transition stage, the soil

starts to desaturate and the water content in the soil reduces significantly with increasing

suction. Thus, there is a nonlinear increase in the modulus of elasticity associated with

increasing soil suction. The amount of water at the soil particle reduces as desaturation

continues. The water menisci area in contact with the soil particles is not continuous and

starts reducing. Eventually, large increases in suction lead to a relatively small change in

water content (or degree of saturation). This stage is referred to as the residual stage of

unsaturation. The suction in the soil at the commencement of this stage is generally

referred to as the residual suction. Beyond the residual suction conditions and during

further desaturation, the soil modulus of elasticity may increase, decrease, or remain

relatively constant. In soils that desaturate relatively fast (e.g., sands and silts), it can be

expected that there is little water left in soil pores when the soil reaches the residual state

and the modulus of elasticity will decrease. The water content in sands and silts at

residual suction conditions can be quite low and may not transmit suction effectively to

the soil particle contact points; therefore, even large increases in suction will not result in

a significant increase in the modulus of elasticity. In contrast, clays may not have a well-

defined residual state. Even at high values of suction there could still be considerable

water available (i.e., in the form of adsorbed water) to transmit suction along the soil

particles, which contributes towards an increase in the modulus of elasticity.

CHAPTER 3 86

Vanapalli and Oh (2010) used the semi-empirical model (Equation 3.1) for estimating the

modulus of elasticity for fine-grained soils with Ip values lower than 16%. The model

was developed based on the results of model footing and in-situ plate load tests available

in the literature. The fitting parameter β = 2 can be used for fine-grained soils, regardless

of the plasticity index Ip value. The fitting parameter α was estimated based on a

relationship proposed between the inverse of α (i.e., 1/α) and the soil plasticity index Ip

as shown in Figure (3.2). The relationship shows that (1/α) non-linearly increases with

increasing Ip. The upper boundary relationship (Equation 3.3) was proposed for soils with

Ip less than 12% and with low suction values. However, the lower boundary relationship

(Equation 3.4) was proposed for soils with Ip less than 16% and with high suction values.

( )02(1/ ) 0.5 0.312( ) 0.109( ) 12p p pI I Iα = + + ≤ ≤ (3.3)

( )02(1/ ) 0.5 0.063( ) 0.036( ) 16p p PI I Iα = + + ≤ ≤ (3.4)

For soils with higher plasticity index (i.e., Ip > 16%), such as the expansive soils, the

validity of Vanapalli and Oh (2010) model (hereafter referred as VO model) as well as its

fitting parameters (i.e. α and β ) values are not available.

Fig. 3.2. Relationship between 1/α and plasticity index Ip (modified after Vanapalli and Oh 2010)

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 87

In this chapter, the results of triaxial tests from the literature for three compacted

expansive soils (i.e., Zao-Yang, Nanyang, and Guangxi) from China were used for

examining the validity of the VO model (Equation 3.1) for unsaturated expansive soils.

The elasticity moduli of soils were determined from the stress-strain curves of triaxial

tests during shearing of saturated/unsaturated compacted specimens under different

confining stresses and matric suctions.

Typically, stress-strain data of triaxial shear test can be represented mathematically by a

hyperbola having the form below (Duncan and Chang 1970) (Figure 3.3).

1 3

1 3

( ) 1( )uE

εσ σ εσ σ

− =+

−

(3.5)

where ε is axial strain, E is initial tangent modulus of elasticity, 1σ and 3σ are major

and minor principal stresses, respectively, and 1 3( )− uσ σ is ultimate deviator stress at a

large strain. The hyperbola is considered valid up to the actual soil failure (i.e., the actual

failure deviator stress 1 3( ) fσ σ− ) (point A as shown in Figure (3.3)). The value of the

parameters of the hyperbolic model (Equation 3.5) E and 1 3( )− uσ σ can be determined

by plotting the stress versus strain relationships on the transformed axes of (axial

strain/deviator stress) 1 3/ ( )−ε σ σ and axial strain ε as shown in Figure (3.4), and

represented by a straight line having the form below (Duncan and Chang 1970).

1 3 1 3

1( ) ( )uE

ε εσ σ σ σ

= +− −

(3.6)

The intercept and the slope of the resulting straight line are the inverse of the initial

tangent modulus of elasticity 1 E and the inverse of ultimate deviator stress 1 31 ( )uσ σ− ,

respectively (Duncan and Chang 1970).

CHAPTER 3 88

Fig. 3.3. Comparison of typical stress-strain curve with hyperbolic stress-strain curve (modified after Al-Shayea et al. 2001)

Fig. 3.4. Transformed hyperbolic stress-strain curve (modified after Duncan and Chang 1970)

The experimental values of modulus of elasticity in this study were determined as the

reciprocal of intercept of the resulting straight lines. However, the predicted values of

elasticity moduli were estimated using the VO model (Equation 3.1). The information

required for using the VO model includes the SWCC and the modulus of elasticity of soil

under the saturated condition Esat along with the two fitting parameters α and β.

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 89

Comparisons were provided between the values of modulus of elasticity derived from the

triaxial tests results and the VO model for the three expansive soils to check its validity.

3.3 Triaxial Shear Test Results and Soils Properties

The experimental results of triaxial shear tests carried out by Zhan (2003), Miao et al.

(2002), and Miao et al. (2007) on compacted expansive soils from Zao-Yang, Nanyang,

and Guangxi, respectively, were used for examining the validity of the VO model

(Equation 3.1) for estimating the modulus of elasticity for unsaturated expansive soils.

Figure (3.5) shows SWCCs for the three expansive soils used in this study. The key

physical properties of these soils are summarized in Table (3.1).

Fig. 3.5. Soil-water characteristic curves (SWCCs) for the three investigated expansive soils (Zao-Yang, Nanyang, and Guangxi expansive soils)

CHAPTER 3 90

Table 3.1. Soil properties of Zao-Yang, Nanyang, and Guangxi expansive soils

Soil type Plasticity index

Liquid limit

Plastic limit

Specific gravity

Dry unit weight

Free swelling

Ip % wL % wp % Gs kN/m3 %

Zao-Yang Zhan (2003)

31.0 50.5 19.5 2.67 15.30 -

Nanyang Miao et al. (2002)

31.8 58.3 26.5 2.7 14.72 74

Guangxi Miao et al. (2007)

31.1 61.4 30.3 2.7 14.52 45

Zhan (2003) carried out conventional and suction-controlled triaxial tests on saturated

and unsaturated compacted soil specimens to investigate the shear strength behavior of

expansive soils. The soil samples were collected from Zao-Yang at about 400 km from

Wuhan in China. The soil has 3% sand, 58% silt, and 39% clay, which can be classified

as silty clay with intermediate plasticity (Zhan 2003). The SWCC for specimens of

compacted Zao-Yang soils shown in Figure (3.5) was measured using a pressure plate

extractor equipped with a 5 bar (1 bar =100 kPa) ceramic stone. The air-entry value of the

soil is about 25 kPa. To represent the in situ state and obtain the average value of the in

situ dry unit weight of 15.3 kN/m3, the compacted specimens were prepared using a static

compaction pressure of 800 kPa on the dry side of optimum at initial water content of

18%. Figure (3.6) summarizes the relationships between the deviator stress 1 3( )−σ σ and

the axial strain ε during shearing of saturated compacted specimens at confining stresses

3σ of 50, 100, 200, and 400 kPa. Figure (3.7) shows the results of the series of triaxial

shear tests for unsaturated compacted specimens under two net confining stresses 3( )au−σ

of 50 kPa and 200 kPa and various matric suction ( )a wu u− values of 25, 50, 100, and 200

kPa. Review of Figure (3.7a) shows an overlap in the stress versus strain relationships for

the specimens tested with ( )a wu u− of 25 kPa and 50 kPa. This may be attributed to some

minor differences in the initial conditions of tested specimens. Also, the low values of

matric suction (25 kPa and 50 kPa) typically have limited influence on the stress-strain

relationship compared to the other higher values of matric suction (100 kPa and 200 kPa).

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 91

Fig. 3.6. Stress-strain curves for specimens of saturated compacted Zao-Yang soils at various confining stresses (modified after Zhan 2003)

(a) Net confining stress = 50 kPa

CHAPTER 3 92

(b) Net confining stress = 200 kPa

Fig. 3.7. Stress-strain curves for specimens of unsaturated compacted Zao-Yang soils: (a) at net confining stress of 50 kPa, (b) at net confining stress of 200 kPa (modified after Zhan 2003)

Miao et al. (2002) studied the shear strength behavior of Nanyang expansive soils under

saturated and unsaturated conditions from triaxial shear tests. The specimens were

remoulded and prepared at a predetermined water content of 17% and unit weight of

14.72 kN/m3 by the static compaction. The SWCC of the Nanyang expansive soils shown

in Figure (3.5) was measured using a pressure plate with a 15 bar ceramic stone. The air-

entry value and the residual suction value of the Nanyang expansive soil were 25 and

1500 kPa, respectively.

Figure (3.8) shows the conventional triaxial tests results of Nanyang soil specimens at the

saturated condition under different confining stresses of 50, 100, and 150 kPa. The

triaxial tests of unsaturated Nanyang soils were performed by controlling matric suction

extending axis translation technique under different net confining stresses. Figure (3.9)

shows the stress-strain curves for the unsaturated soil specimens with initial matric

suctions of 50, 80, 120, and 200 kPa.

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 93

Fig. 3.8. Stress-strain curves for specimens of saturated compacted Nanyang soils (modified after Miao et al. 2002)

(a) Matric suction, ( )a wu u− = 50 kPa

CHAPTER 3 94

(b) Matric suction, ( )a wu u− = 80 kPa

(c) Matric suction, ( )a wu u− = 120 kPa

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 95

(d) Matric suction, ( )a wu u− = 200 kPa

Fig. 3.9. Stress-strain curves for specimens of unsaturated compacted Nanyang soils: (a) at matric suction of 50 kPa, (b) at matric suction of 80 kPa, (c) at matric suction of 120 kPa, and (d) at matric suction of 200 kPa (modified after Miao et al. 2002)

Miao et al. (2007) also carried out triaxial shear tests to study the mechanical behavior of

Guangxi soils for varying degrees of saturation (i.e., different suctions). The focus of

their study was directed to understand the stress-strain-volume change behavior at

different values of the degree of saturation. The specimens were statically compacted in a

brass cylinder into five layers to achieve a dry unit weight of 14.52 kN/m3. The

compacted soil specimens were prepared at different initial degrees of saturation (i.e.,

76.3%, 83.5%, 92.1%, and 100%). The degree of saturation was controlled by the initial

water content of the soil specimens. The specimens might have some differences in the

soil structure as a result of the compaction at different water contents to a certain density

(Miao et al. 2007). The SWCC of the compacted Guangxi soils is shown in Figure (3.5).

The air-entry value of the compacted Guangxi soils is about 30 kPa.

The conventional triaxial tests were used to study the stress-strain behavior of the

saturated specimens of Guangxi soils under confining stresses of 50, 100, and 200 kPa

(Figure 3.10). The triaxial tests under drained conditions were performed on specimens

under unsaturated conditions. Figure (3.11) shows the stress-strain curves for Guangxi

CHAPTER 3 96

soils specimens tested under unsaturated conditions with varying initial degrees of

saturation (i.e., 76.3, 83.5, and 92.1%) and net confining stresses (i.e., 50, 100, and 200

kPa).

Fig. 3.10. Stress-strain curves for specimens of saturated compacted Guangxi soils (modified after Miao et al. 2007)

(a) Degree of saturation, S = 76.3%

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 97

(b) Degree of saturation, S = 83.5%

(c) Degree of saturation, S = 92.1%

Fig. 3.11. Stress-strain curves for specimens of unsaturated compacted Guangxi soils: (a) at degree of saturation of 76.3%, (b) at degree of saturation of 83.5%, and (c) at degree of saturation of 92.1% (modified after Miao et al. 2007)

3.4 Analysis of the Triaxial Tests Results

The stress-strain curves of the triaxial tests for the three expansive soils (i.e., Zao-Yang,

Nanyang, and Guangxi soils) were analyzed and plotted on the transformed axes

1 3/ ( )ε σ σ− and ε as suggested by Duncan and Chang (1970). The straight line equation

CHAPTER 3 98

(Equation 3.6) was used to fit the data. The experimental values of modulus of elasticity

were determined as the reciprocal of the intercepts of the straight lines. Figures (3.12) and

(3.13) show the transformed stress-strain curves for saturated and unsaturated Zao-Yang

soils specimens, respectively. The values of experimental elasticity moduli determined as

the reciprocal of the intercepts of the straight lines are also shown in the Figures (3.12)

and (3.13).

Fig. 3.12. Transformed stress-strain curve for specimens of saturated, compacted Zao-Yang soils

(a) Net confining stress = 50 kPa

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 99

(b) Net confining stress = 200 kPa Fig. 3.13. Transformed stress-strain curves for specimens of unsaturated, compacted Zao-Yang soils: (a) at net confining stress of 50 kPa, (b) at net confining stress of 200 kPa

Review of Figure (3.12) shows that the experimental value of the saturated modulus of

elasticity increases with an increase in the applied confining stress. This is consistent with

Janbu’s relationship (1963) (Equation 3.7), in which the soil modulus of elasticity under

the saturated condition is related to the confining stress, and is nonlinearly increased with

an increase in the confining stress.

3( )na

a

E K PPσ

= (3.7)

where aP is atmospheric pressure expressed in the same stress units as E and 3σ , K and

n are fitting parameters. The Janbu’s relationship (1963) (Equation 3.7) was used to fit

the data of specimens of saturated, compacted Zao-Yang soils (Figure 3.14). The

coefficient of determination was relatively high (R2 = 0.97). Hence, the modulus of

elasticity of saturated, compacted Zao-Yang soils at any confining stress can be estimated

using equation (3.8).

0.67353.29( )a a

EP P

σ= (3.8)

CHAPTER 3 100

Fig. 3.14. The relationship of the saturated modulus of elasticity with the confining stress for specimens of compacted Zao-Yang soils

On the other hand, Figure (3.13) shows that the soil modulus of elasticity under an

unsaturated condition increases with an increase in the matric suction ( )a wu u− . This is

consistent with the observations of other investigators in the literature (Fredlund and

Rahardjo 1993, Costa et al. 2003, Inci et al. 2003, Lu and Likos 2006, Yang et al. 2005,

Yang et al. 2008, Oh et al. 2009, Vanapalli and Oh 2010). Review of Figure (3.13) shows

that some sets of data (e.g., the transformed stress-strain relationship for 3( )− auσ = 200

kPa and ( )−a wu u = 25 kPa) are not linear; such a behavior can be attributed to the stress-

strain curve not following well defined hyperbolic trend with respect to its shape.

Nonetheless, a straight line could be still fitted to the data. This assumption is reasonably

valid as the coefficient of determination was relatively high (R2 > 0.96). Table (3.2)

summarizes the experimental values of the modulus of elasticity for Zao-Yang soils

under the saturated and unsaturated conditions.

The above procedure was also applied for the other two soils (Nanyang and Guangxi

soils) to determine their experimental elasticity moduli. Tables (3.3) and (3.4) summarize

the experimental values of elasticity modulus for Nanyang and Guangxi soils,

respectively.

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 101

Table 3.2. Experimental elasticity moduli obtained from the triaxial tests for compacted

Zao-Yang soils under the saturated and unsaturated conditions (data from Zhan 2003)

Net confining stress (kPa)

Saturated modulus of elasticity (kPa)

Matric suction (kPa) 25 50 100 200 Unsaturated modulus of elasticity (kPa)

50 3333 20000 33333 50000 50000 100 5000 - - - - 200 10000 12700 - 25000 33333 400 12500 - - - -

Table 3.3. Experimental elasticity moduli obtained from the triaxial tests for compacted

Nanyang soils under the saturated and unsaturated conditions (data from Miao et al.

2002)

Net confining stress (kPa)

Average confining stress (kPa)

Saturated modulus of elasticity (kPa)

Matric suction (kPa) 50 80 120 200 Unsaturated modulus of elasticity (kPa)

20 25 9253* - 25000 - 50000 30 12921* 25000 - 33333 - 50 62.5 20000 33333 - - 50000 70 25964* - 33333 - - 80 28982* - - 50000 - 100 112.5 33333 50000 - - 50000 120 40473* - 50000 - - 130 43231* - - 50000 - 150 - 50000 - - - -

* The value obtained from the best fit of the relationship between the saturated modulus of elasticity and the confining stress using Janbu’s equation (1963) (Equation 3.7, the fitting parameters K = 347.47, and n = 0.82)

Table 3.4. Experimental elasticity moduli obtained from the triaxial tests for compacted

Guangxi soils under the saturated and unsaturated conditions (data from Miao et al. 2007)

Net confining stress (kPa)

Saturated modulus of elasticity (kPa)

Degree of saturation (%) 76.3 83.5 92.1 Unsaturated modulus of elasticity (kPa)

50 10000 16667 20000 10000 100 10000 20000 20000 12500 200 12500 25000 25000 20000

CHAPTER 3 102

Figure (3.9) and Table (3.3) summarize Miao et al. (2002) results of 12 triaxial tests for

specimens of Nanyang soils under unsaturated conditions using different values of

confining stresses and matric suctions. However, to validate Vanapalli and Oh (2010)

model for estimating the modulus of elasticity of unsaturated expansive soils, at least

three specimens have to be tested under the same confining stress with varying initial

matric suctions. Therefore, tests conducted by Miao et al. (2002) for specimens under

unsaturated conditions were divided into three groups. Every group was consisted of four

tests having a close net confining stress. The net confining stress for each group was

determined as an average value of the net confining stresses of the four tests in the group.

Table (3.3) shows the confining stresses for the three groups used in this analysis, which

are 25, 62.5, and 112.5 kPa.

3.5 Comparison between the Experimental and Predicted Values of the

Modulus of Elasticity

The predicted moduli of elasticity of Zao-Yang, Nanyang, and Guangxi expansive soils

were calculated using the VO model (Equation 3.1). This VO model requires the SWCC

and the modulus of elasticity under the saturated condition along with the fitting

parameters α and β. The saturated modulus of elasticity was determined from the stress-

strain curves of the soil specimens tested under the saturated condition. The fitting

parameter β = 2 was used for expansive soils regardless of the value of the plasticity

index following the VO model recommendation for fine-grained soils. It was also found

that β = 2 provides a reasonable estimation of the variations of soil movements with

respect to time for several case studies with lightly loaded structures as will be presented

later in Chapter Five. The value of the fitting parameter α was defined using a

programming code for equation (3.1) that increments the value of α and calculates the

predicted modulus of elasticity of unsaturated soils. The value of α was defined on the

basis of the best agreement between the experimental and the predicted values of the

modulus of elasticity with respect to matric suction (the coefficients of determination R2

= 0.77−0.97). Tables (3.5)−(3.7) summarize the predicted elasticity moduli and the

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 103

corresponding values of the fitting parameters α and β for Zao-Yang, Nanyang, and

Guangxi expansive soils, respectively.

Table 3.5. The fitting parameters and the predicted elasticity moduli estimated using the

VO model for unsaturated, compacted Zao-Yang soils

Net confining stress (kPa)

Fitting parameter values Matric suction (kPa) 1/α α β 25 50 100 200

Predicted modulus of elasticity (kPa) 50 7.4 0.135 2 14383 22526 36385 62991 200 50 0.02 2 14911 - 24690 36515

Table 3.6. The fitting parameters and the predicted elasticity moduli estimated using the

VO model for unsaturated, compacted Nanyang soils

Average net confining stress (kPa)

Fitting parameter values Matric suction (kPa) 1/α α β 50 80 120 200

Predicted modulus of elasticity (kPa) 25 25 0.04 2 22215 27454 33102 44218 62.5 58.82 0.017 2 33680 38415 43521 53568 112.5 166.67 0.006 2 44122 46834 49758 55512

Table 3.7. The fitting parameters and the predicted elasticity moduli estimated using the

VO model for unsaturated, compacted Guangxi soils

Net confining stress (kPa)

Fitting parameter values Degree of saturation (%) 1/α α β 76.3 83.5 92.1

Unsaturated modulus of elasticity (kPa) 50 14.29 0.07 2 17214 15867 13362 100 100 0.01 2 20306 18382 14802 200 100 0.01 2 25382 22978 18503

Tables (3.5)−(3.7) show that the predicted value of the soil modulus of elasticity

increases with an increase in matric suction ( )a wu u− and net confining stress 3( )au−σ

with the exception of the Zao-Yang soils specimens tested by Zhan (2003) under the net

confining stress of 50 kPa having a higher elastic modulus than the specimens tested

under the confining stress of 200 kPa. This exception appears to be inconsistent with the

CHAPTER 3 104

remainder of the results, and may be attributed to the effect of dilatancy of the soil at a

low confining stress (i.e. 3( )auσ − = 50 kPa). During dilation, a negative pore water

pressure can develop and the soil stiffness (modulus of elasticity) increases. However, the

effect of dilation is negligible at high confining stresses (e.g., 3( )auσ − = 200 kPa). These

results are consistent with the observations of other investigators in the literature (Zhan

2003, Vanapalli and Mohamed 2013).

Figures (3.15)−(3.17) provide the comparisons between the experimental and the

predicted moduli of elasticity for Zao-Yang, Nanyang, and Guangxi soils. The results

show that the soil modulus of elasticity increases with an increase in the matric suction

( )a wu u− and the net confining stress 3( )au−σ with the exception of the Zao-Yang soil

data as it has been discussed above. In addition, a reasonable agreement has been

observed between the experimental and the predicted values of the elasticity moduli for

all the three expansive soils studied (R2 = 0.91) (Figure 3.18). This close agreement

suggests the use of the VO model (Equation 3.1) with confidence for estimating the

modulus of elasticity for unsaturated expansive soils.

Fig. 3.15. Comparison between the experimental and predicted modulus of elasticity for Zao-Yang expansive soils

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 105

Fig. 3.16. Comparison between the experimental and predicted modulus of elasticity for Nanyang expansive soils

Fig. 3.17. Comparison between the experimental and predicted modulus of elasticity for Guangxi expansive soils

CHAPTER 3 106

Fig. 3.18. Predicted moduli versus experimental moduli for the three investigated expansive soils (Zao-Yang, Nanyang, and Guangxi expansive soils)

3.6 The Relationship between the Plasticity Index IP, the Net Confining

Stress 3( )auσ − , and the Fitting Parameter α

The values of the fitting parameter α for the three expansive soils studied, summarized

in Tables (3.5)−(3.7), were plotted along with the upper and lower boundary relationships

of (1/α) versus Ip proposed by Vanapalli and Oh (2010) for soils with plasticity index Ip

lower than 16% (Figure 3.19). Review of Figure (3.19) shows that the inverse of the

fitting parameter α for soils under low confining stresses are below the lower boundary

relationship, and that for high confining stresses are in the range of the boundary

relationships. However, there is an exception for the specimens tested by Miao et al.

(2002) that have a high value of (1/α) when tested under the average net confining stress

of 112.5 kPa.

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 107

Fig. 3.19. The plot of (1/α) versus the plasticity index Ip for the three investigated expansive soils along with the upper and lower boundary relationships of (1/α) versus Ip proposed by Vanapalli and Oh (2010)

Vanapalli and Oh (2010) suggested that the fitting parameter α is a function of footing

size and soil plasticity index Ip. Figure (3.19) shows that the value of α is also dependent

on the applied net confining stress. For a soil with a certain value of Ip, the fitting

parameter α may differ with the applied confining stress. Figure (3.20) shows the values

of 1/α with the associated net confining stresses for the three expansive soils under

consideration. These results suggest that 1/α may nonlinearly increase with an increase in

the applied net confining stress. It is also found that the value of α between 0.05 and 0.15

with an average 0.1 provides a reasonable estimation of the modulus of elasticity for

unsaturated expansive soils under a net confining stress of less than 50 kPa. This is

typically the maximum overburden pressure (or the confining stress) of the soil deposit

within the active zone depth (approximately 3 m) where the volume change problems of

expansive soils are predominant. The values of the modulus of elasticity of unsaturated

expansive soils estimated using the fitting parameters β = 2 and α = 0.05-0.15 provide

reasonable predictions of the soil vertical movements with respect to time for the case

studies investigated in this study (these details are presented in Chapter Five). In other

words, the results of the analyses summarized in this chapter with respect to the values of

the two fitting parameters (i.e., α and β ) for reasonably estimating the modulus of

elasticity of unsaturated expansive soils are consistent with the assumptions used in

CHAPTER 3 108

Chapter Five for α and β values for predicting the vertical movements associated with

the volume change behavior of expansive soils. This approach provides conservative

estimations as the boundary restraints due to loading on the swelling behavior are not

considered. Such an approach is simple for using in practice applications.

Some investigators suggest that the active zone depth for some sites may extend up to 20

ft. (∼ 6 m) below the ground surface (Kalantari 2012). Nelson et al. (2001) showed, due

to both soil suction and gravity, wetting extend to depths much greater than 6.1 m at sites

in Denver. Diewald (2003) evaluated post-construction data from 133 investigations and

determined that the depth of wetting for 7 to 10 year-old residences is approximately 12

m. Some practicing engineers in the Front Range of Colorado have used assumptions of

depths of wetting of 10.4−14.0 m for their predictions of soil heave (Chao et al. 2006).

For any site with deep active zone (i.e., > 3 m), the relationships between 1/α and the net

confining stress values should be established and used for a reliable estimation of soil

movements using the modulus of elasticity as a tool.

Fig. 3.20. The plot of (1/α) versus net confining stress 3( )auσ − for the three investigated expansive soils (Zao-Yang, Nanyang, and Guangxi expansive soils)

MODULUS OF ELASTICITY OF UNSATURATED EXPANSIVE SOILS 109

3.7 Summary

Vanapalli and Oh (2010) proposed a semi-empirical model (i.e., VO model) for

predicting the variation of the modulus of elasticity with respect to matric suction for

soils with plasticity index Ip lower than 16%. The model has been extended in this

chapter to predict the modulus of elasticity of unsaturated expansive soils which,

typically, have higher plasticity index. The information required for using the VO model

includes the SWCC and the modulus of elasticity of the soil under the saturated condition

along with two fitting parameters α and β. The experimental data of triaxial tests

available in the literature for three different expansive soils were used to examine the

validity of the adopted model.

The results of the study suggest that the VO model can be used for predicting the

variation of the modulus of elasticity with respect to matric suction for unsaturated,

compacted expansive soils (e.g., Zao-Yang, Nanyang, and Guangxi expansive soils) ( the

coefficient of determination R2 = 0.91). The fitting parameter β = 2 was found to be

suitable for the three expansive soils studied in this chapter. The fitting parameter α is

related to the net confining stress 3( )au−σ . However, to provide a generalized

relationship for the fitting parameter α, more triaxial test results for different expansive

soils with different plasticity index Ip tested under a large range of confining stresses and

matric suctions are required. The results presented in this chapter are encouraging for use

of the VO model in the modeling studies to reasonably predict the variation of soil

movements with respect to time (see Chapter Five).

CHAPTER 3 110

CHAPTER 4

PROPOSED APPROACH FOR PREDICTING VERTICAL

MOVEMENTS OF EXPANSIVE SOILS

4.1 Introduction

Significant research has been undertaken since the last century to better understand the

heave/shrink behavior of expansive soils. Terzaghi’s (1925, 1926, and 1931) pioneering

modeling and experimental studies showed that clay swelling and shrinkage are

essentially elastic deformations caused by the clay’s affinity for water. Terzaghi (1926)

investigated the mechanics of the swelling of a gelatin gel, as a model for clay, using the

thermodynamic principles. Empirical relationships were proposed considering many

parameters that influence the swelling behavior of gel such as the concentration of gel,

the size of gel micropores, and the temperature. The swelling pressure of the gel was

found to be merely due to the elastic expansion of the solid phase, previously held under

compression by the surface tension of the water. The swelling pressure represents the

“free energy” of the system and can be entirely converted into mechanical work. The heat

developed in connection with a change in the free energy with unrestricted expansion is

exclusively due to liquid friction as the gel expands. In addition, Terzaghi (1931)

explained the fundamental swelling behavior of a two-phase system of liquid and solid

(i.e., water and soil) using two different scenarios as examples. In the first scenario, the

initial water content w0 of the submerged two-phase system was reduced to a value w1 by

applying external pressure p per unit area. The volume was reduced by ′ ′−aa bb (Figure

4.1a). In the second scenario, the water content was reduced to w1 by drying instead of

applying the external pressure p (Figure 4.1b); assuming that no air has invaded into the

system (i.e., two-phase system). The volume reduction in first scenario was due to

compression by load (i.e., consolidation), and it was associated with shrinkage by

111

evaporation in the second scenario. When the applied external pressure p was removed in

the first scenario, water flowed into the system accompanied by swelling. To initiate

swelling in the second scenario, the surface ′ ′−a b of the system represented in Figure

(4.1b) was flooded with water. In both scenarios, swelling started with identical water

content w1 and particles arrangement. In the first scenario, a load of p per unit area of

surface was necessary to prevent the material from expanding. In the second scenario,

there must be a force of equal intensity (as in the first scenario) acting on the surface

′ ′−a b in the system. This force is exerted by the surface tension of the capillary water

(i.e., suction). The difference between the pressure in the external water (free water) and

in the interstitial water causes water to flow through the surface into the two-phase

system until the influence of hydraulic gradient ceases. In other words, the system

expands until the suction becomes zero. The water flow is independent of the

physiochemical reaction inside the two-phase system. This leads to a conclusion that the

physiochemical effect could not have had any influence on the swelling behavior of the

system.

Terzaghi (1931) suggested that the most important factor that contributes to swelling is

the negative pressure (suction) associated with the capillary water in the interconnected

pores of the clay macrostructure. Terzaghi (1925, 1931) also pointed out that the soil

swelling produced by eliminating the surface tension of the capillary water (suction) was

identical with the expansion produced by the removal of the external load. It was

explained that any soil that is capable of swelling contains a solid phase under a pressure

equal to the tension in the liquid phase. Hence, the swelling capacity of soil is dependent

on the elastic properties of the solid phase of the soil. This is a fundamental concept and

can be extended to all expansive soils. However, the physiochemical reactions between

the solid and the liquid phases (formation of adsorption compounds within the system)

can at best only play a minor role.

CHAPTER 4 112

Fig. 4.1. Capillary pressure and swelling process (modified after Terzaghi 1931)

As per Terzaghi’s (1925, 1926, and 1931) explanation of the swelling process of soil, a

simple approach for predicting the time-dependent soil movements (heave/shrink) of

natural expansive soils beneath lightly loaded structures is proposed in this research

study. The proposed approach depends on the variations of the soil modulus of elasticity

with respect to matric suction; it is therefore referred to as the modulus of elasticity based

method (MEBM). The matric suction variations within the active zone of soil profile are

simulated using the soil-atmosphere interaction model VADOSE/W. The soil vertical

movements (heave/shrink) over time are estimated based on the volume change

constitutive relationship for soil structure considering the variation of modulus of

elasticity with respect to matric suction. This chapter details the assumptions and the

fundamental concepts of the proposed MEBM along with the step-by-step procedural

details for predicting the time-dependent movement of natural expansive soils.

(a) Reducing the water content by applying external load

(b) Reducing the water content by drying

Initial condition Initial condition

PROPOSED APPROACH FOR PREDICTING VERTICAL MOVEMENTS OF EXPANSIVE SOILS 113

4.2 Constitutive Relationship for Estimating Expansive Soil Movements

over Time

Fredlund and Morgenstern (1976) proposed two volume change constitutive relationships

as shown in equations (2.14) and (2.15) for soil structure and water phase, respectively. A

total of four volume change coefficients ( 1sm , 2

sm , 1wm , and 2

wm ) were defined to

completely describe the volume-mass soil properties under any set of stress conditions

(Vu and Fredlund 2004).

1 20

( ) ( )ε σ= = − + −s svv mean a a w

dVd m d u m d u uV

(2.14)

1 20

( ) ( )w www mean a a w

dVd m d u m d u uV

θ σ= = − + − (2.15)

where vε is volumetric strain, wθ is volumetric water content, 0( )vdV V is total volume

change, 0( )wdV V is water volume change, ( )mean auσ − is net normal stress, ( )a wu u−

matric suction, 1sm is coefficient of volume change with respect to net normal stress, 2

sm

is coefficient of volume change with respect to matric suction, 1wm is coefficient of water

volume change with respect to net normal stress, and 2wm is coefficient of water volume

change with respect to matric suction. However, when predicting soil deformation, a

change in soil volumetric strain is of main interest. Therefore, only the constitutive

relationship of volume change for soil structure (Equation 2.14) is required (Fredlund et

al. 1980).

Following Terzaghi (1925, 1926, 1931), and other researchers (e.g., Biot 1941, Coleman

1962, Matyas and Radhakrishna 1968, Barden et al. 1969, Aitchison and Woodburn

1969, Brackley 1971, Aitchison and Martin 1973, Fredlund and Morgenstern 1976 and

1977, Vu and Fredlund 2004 and 2006, Zhang 2004, Zhang and Briaud 2010), a simple

method for predicting the vertical movements of expansive soils over time is proposed in

this study based on an assumption that expansive soils are elastic in nature for a large

range of loading conditions. This assumption is considered valid since it is reasonable to

CHAPTER 4 114

assume that any unsaturated expansive soil has experienced the maximum wetness and

dryness in the past. In other words, if an expansive soil has some plasticity, its

contribution could have been eliminated by a long history of wetting-drying cycles. This

may be one of the key reasons why most expansive soils are usually heavily

overconsolidated. Since the volume change behavior of expansive soil is influenced by

the mechanical stress, one may argue that the soil will yield under a combination of

mechanical stress and matric suction variations. However, for pavements and lightly

loaded structures, where the proposed prediction method (MEBM) can be used, the

mechanical stress due to the repeated traffic or the superstructure loading is not

significant and will not cause soil yielding (Fredlund et al. 1980).

Assuming soils behave as incrementally isotropic, linear elastic materials, the volume

change coefficients 1sm , and 2

sm can be related to the elastic moduli E and H associated

with a change in the net normal stress and a change in the matric suction, respectively.

For an unsaturated soil under a general, three-dimensional loading condition,

1 3(1 2 )sm Eµ= − , and 2 3sm H= , where µ is Poisson’s ratio; thus, the soil structure

constitutive relationship can be rewritten as equation (2.12) (Fredlund and Rahardjo

1993)

(1 2 ) 33( ) ( - ) ( - )v mean a a wd d u d u uE H

µε σ−= + (2.12)

If the soil is subjected to an increase in the matric suction, the soil volume will be the

same as long as the soil remains saturated. Once the soil commences desaturation, the

changes in the matric suction and the mechanical stress will affect the volume change

behavior. However, as mentioned above, the influence of the mechanical stress in lightly

loaded structures is insignificant in several scenarios and can be neglected. Such an

assumption is conservative and can be extended in practice. In other words, the matric

suction can be regarded as the only key stress state variable contributing to the soil

volume change. The constitutive relationship for the soil structure (Equation 2.12)

reduces to equation (4.1)

PROPOSED APPROACH FOR PREDICTING VERTICAL MOVEMENTS OF EXPANSIVE SOILS 115

3 ( )v a wd d u uH

ε = − (4.1)

Equation (4.1) suggests that the matric suction changes have a direct bearing on the

volume change of unsaturated expansive soils as per the earlier discussions on the

swelling process provided by Terzaghi (1925, 1926, 1931). The soil swelling merely

represents elastic expansion produced by lowering of the capillary pressure (suction)

(Terzaghi 1925, 1926, 1931).

The movements of expansive soils associated with the changes in environmental factors

often occur near the ground surface within the active zone depth. Hence, the soil

movements may be assumed to be predominant in the vertical direction, and the loading

condition can be assumed to be the K0-loading. The volumetric strain vdε is equal to

the vertical strain ydε while the soil is not permitted to deform laterally (i.e., xdε = zdε

= 0). The volume change coefficient with respect to matric suction is

2 (1 ) / ( (1 ))sm Hµ µ= + − . Equation (4.1) can therefore be written in terms of the

vertical strain as follows

(1 ) ( )(1 )y a wd d u u

Hµεµ

+= −

− (4.2)

The elastic moduli H and E of unsaturated soils vary significantly with the stress state

variables (i.e., the mechanical stress and the matric suction), and the elastic modulus

H can be related to E and µ (Wong et al. 1998, Zhang et al. 2012) as

( )/ 1 2H E= − µ (4.3)

Equation (4.3) has two unknowns (H and E) since Poisson’s ratio µ is usually

estimated or measured in the laboratory through the use of triaxial tests with the

measurement of lateral strain. The relationship between H and E may be more

complex for soils in a state of unsaturated condition; however, equation (4.3) that is

valid for saturated soils has been applied to unsaturated soils in the present study

CHAPTER 4 116

extending the assumptions suggested by Wong et al. (1998), Zhang et al. (2012), and

Geo-Slope (2007).

Substituting for H (Equation 4.3) into equation (4.2) provides equation (4.4) that relates

the vertical strain in terms of the matric suction change, the associated modulus of

elasticity for the soil structure E, and Poisson’s ratio µ.

(1 )(1 2 ) ( )(1 )y a wd d u u

Eµ µε

µ+ −

= −−

(4.4)

The summation of the vertical strain changes for each an increment provides the final

vertical strain of the soil.

y ydε ε= ∑ (4.5)

To calculate the vertical movement of expansive soils with respect to time, the soil

profile within the active zone are subdivided into several layers. The vertical

movement for each arbitrary layer (i.e., ith layer) ih∆ associated with the time

increment is computed by multiplying the vertical strain yε at the mid-layer for the

time increment with the thickness of the layer ih .

(1 )(1 2 ) ( )(1 )

µ µ∆ ∆µ

+ −= − −

i i a wi

h h u uE

(4.6)

where ( )a wu u∆ − is change in soil suction for each time increment.

The vertical movement of each soil layer at a given time is the cumulative value of the

vertical movement increments prior to that given time. The total vertical movement h∆

at any point in the soil profile is the summation of the vertical movements of n layers

beneath.

1 1

(1 )(1 2 ) ( )(1 )

n n

i i a wi i i

h h h u uE= =

+ µ − µ∆ = ∆ = ∆ − − µ

∑ ∑ (4.7)

PROPOSED APPROACH FOR PREDICTING VERTICAL MOVEMENTS OF EXPANSIVE SOILS 117

Since it is necessary to define a value for Poisson’s ratio, equation (4.7) infers that the

matric suction variations in the active zone and the associated modulus of elasticity

are the key parameters to calculate the vertical soil movements (heave/shrink) over

time. The matric suction variations within the active zone are simulated using the soil-

atmosphere interaction model VADOSE/W, while the corresponding values of

unsaturated modulus of elasticity are calculated using the VO model (Equation 3.1).

4.3 Key Parameters for Predicting the Expansive Soil Movements

4.3.1 Matric suction variations

The matric suction profile within the active zone is a representation of a state of balance

of the environmental factors and the soil-water storage processes. The potential change in

matric suction is generally attributed to many reasons such as the environmental

conditions, human imposed irrigation, influence of vegetation, and accidental wetting due

to broken pipelines. Since even small changes in soil suction may cause a significant

amount of volume change, the soil suction has been considered as a more sensitive

indicator and a predominant stress variable for the determination of soil movements.

To estimate the soil vertical movements over time using the proposed MEBM

(Equation 4.7), the in-situ matric suction changes and the expected depth of these

changes (i.e., the active zone depth) due to the variations of environmental factors are

required. Several commercial programs such as SoilCover (Unsaturated Soils Group

1996), HYDRUS-2D (Simunek et al. 1999), UNSAT-H (Fayer 2000), VADOSE/W

(Geo-Slope 2007), and SVFlux (SoilVision System Ltd. 2007) are available for

estimating the matric suction profiles considering both the water flow in unsaturated soils

and the soil-atmospheric interactions (i.e., infiltration, precipitation, surface water

runoff and ponding, plant transpiration, actual evaporation, and heat flow). Such

techniques are valuable, simple, and economical in comparison with the direct

measurements of in-situ matric suction which are mostly unreliable.

The finite element program VADOSE/W (Geo-Slope 2007), a product of Geo-studio, is

used in this research study as a tool to simulate the soil-atmospheric interactions and the

CHAPTER 4 118

water flow through unsaturated soils, and then to estimate the corresponding changes in

matric suction over time. The program couples the heat transport and mass (i.e., water

and vapor) transfer in unsaturated soils, together with the water and energy balance, to

provide a direct and complete evaluation of soil water storage and matric suction. Critical

to the formulation of VADOSE/W is its ability to predict the actual evaporation as a

function of climate data applied as an upper boundary condition using the rigorous

Penman-Wilson method (Wilson 1990). It has been established that VADOSE/W is

capable of simulating the saturated and unsaturated flow behavior where the complex

soil-atmosphere interaction is of particular interest.

VADOSE/W is primarily a 2-D package but can be used for a 1-D problem through the

use of appropriate geometries and boundary conditions. Unsaturated flow problems with

atmospheric interactions are often conducted in 2-D when the soil surface is sloping or

the layering in the profile promotes multi-dimensional flow. The VADOSE/W program

requires the following input information: (i) material properties, namely the soil-water

characteristic curve (SWCC), the coefficient of permeability function (k function), the

soil thermal conductivity, the mass, and the specific heat capacity, (ii) climate data that

includes the daily precipitation, the maximum and minimum daily temperature, the

maximum and minimum daily relative humidity, the average daily wind speed, and the

net radiation, (iii) vegetation data which involves the leaf area index (LAI), the plant

moisture limiting point, the root depth and length in the growing season, and (iv)

geometrical boundary conditions including the location of the ground water table.

For a proper simulation of unsaturated flow, a correct description of the boundary and

initial conditions is important. Different types of boundary conditions are included in

VADOSE/W to simulate various problems. The boundary conditions which are generally

applied at the bottom of the soil profile include unit gradient condition (e.g., gravity

driven flux = the actual hydraulic conductivity at the bottom of the domain), seepage face

(e.g., flux = the saturated hydraulic conductivity when the boundary is saturated;

otherwise flux = 0), prescribed head, or prescribed flux. An atmospheric flux boundary

condition is another type of boundary conditions which is usually applied at the surface to

simulate the atmospheric interactions. Infiltration occurs during precipitation at a rate PROPOSED APPROACH FOR PREDICTING VERTICAL MOVEMENTS OF EXPANSIVE SOILS 119

governed by the hydraulic properties of the soil profile, and precipitation exceeding the

infiltration capacity is assumed to be run off. Evaporation is assumed to occur from the

soil surface and is bounded by the potential evaporation (PE) rate. For 2-D simulations,

the prescribed head and prescribed flux boundaries are often applied along the sides of

the domain. A detailed description of different boundary conditions used in VADOSE/W

is presented in Geo-Slope (2007).

The initial condition is also required for the simulation of the transient water flow

through unsaturated soils, which usually includes the initial values of total head, soil

temperature, and gas concentration at each nodal point within the soil profile. When this

data is not available, a value of zero for the gas concentration and the soil temperature

can be assumed as an initial value. However, the hydraulic initial condition state cannot

be left out, and should be specified by reading the data from a file created in a separate

analysis of the initial condition, by drawing the initial water table position, or by

specifying the initial value as a material property.

The output of VADOSE/W includes the soil temperature, degree of saturation, water

content, and, most importantly, matric suction fluctuations over time. Details of

VADOSE/W simulations for the case studies used in this research are discussed in the

sequent chapter.

4.3.2 Soil modulus of elasticity associated with matric suction

Matric suction may be sufficient as the sole independent variable in describing the

modulus of elasticity for unsaturated soils (Fredlund and Rahardjo 1993, Costa et al.

2003, Inci et al. 2003, Lu and Likos 2006, Yang et al. 2008). The soil modulus of

elasticity is significantly influenced due to the contribution of matric suction. Typically,

as matric suction becomes bigger, the modulus of elasticity becomes higher.

The semi-empirical model presented in Chapter Three (i.e., VO model) (Equation 3.1) is

used in the proposed MEBM for estimating the modulus of elasticity of unsaturated

expansive soils associated with the change in matric suction. The information required

for using the VO model include the SWCC and the saturated modulus of elasticity along

with the two fitting parameters β and α. As described in Chapter Three, the VO model

CHAPTER 4 120

was validated using the experimental data of triaxial tests for three different expansive

soils. Comparisons were provided between the values of the modulus of elasticity

obtained from the triaxial tests and the VO model. The adopted VO model well estimates

the modulus of elasticity of unsaturated expansive soils. The values of the fitting

parameters of β = 2 and α = (0.05-0.15) with an average 0.1 were recommended for

expansive soils. In the MEBM approach, the vertical soil movements (heave/shrink) are

predicted using equation (4.7) considering the variation of modulus of elasticity with

respect to matric suction estimated from the VO model (Equation 3.1). The strength of

the VO model lies in its use of conventional soil properties.

4.4 Step-by-Step Procedure of the Proposed MEBM Approach

Figure (4.2) shows the step-by-step procedure of the MEBM for predicting the vertical

movement of unsaturated expansive soils with respect to time. The proposed approach

requires the soil matric suction variations within the active zone depth and the

corresponding modulus of elasticity. The soil matric suction variations are simulated

using the soil-atmosphere interaction model VADOSE/W. The corresponding values of

the soil modulus of elasticity are estimated using the VO model (Equation 3.1). The soil

movements (heave/shrink) over time are then predicted using equation (4.7).

PROPOSED APPROACH FOR PREDICTING VERTICAL MOVEMENTS OF EXPANSIVE SOILS 121

Fig. 4.2. Flowchart for the step-by-step procedure of the MEBM

To simulate the matric suction changes for each time increment in response to

atmospheric conditions, the investigated soil profile for a given site may be modelled

using the transient analysis with the 2-D VADOSE/W program. Besides the soil

properties, the initial and the boundary conditions of the site model are applied over the

period of simulation. The matric suction variations within the active zone depth can be

predicted as a response to a changing surface boundary over time. Once the matric

suction variations are determined, the vertical soil movements at any depth can be

predicted. The investigated soil profile is divided into a number of sub-layers based on

the soil properties and the considered locations. The total soil movement at a specific

location for a given time is computed by adding the movements of all layers up to the

specific location for that time (Equation 4.7).

Five different case studies for studying the volume change behavior of expansive soils,

from different countries that include Canada, China, and the United States, are used in

this research study for testing the validity of the proposed MEBM (see the sequent

Chapter). Those case studies include:

CHAPTER 4 122

- Case Study A is a slab-on-ground placed on Regina expansive clay subjected to a

constant infiltration rate over 175 days, which was originally modeled by Vu and

Fredlund (2006).

- Case Study B is a typical light industrial building in north-central Regina,

Saskatchewan, Canada, constructed by the Division of Building Research, National

Research Council in 1961. The history of the site and the details of testing and

monitoring programs are presented in Yoshida et al. (1983). The prediction of soil

heave to investigate problems associated with the construction of the building was

carried out by Vu and Fredlund (2004) over 150 days.

- Case Study C is a test site in Regina, Saskatchewan, Canada, modeled by Ito and Hu

(2011) over a 1-year period to investigate the performance of water pipe lines in

Regina expansive clays. Various factors influencing the soil movements such as

climate condition, vegetation, watering of lawn and structural impact are considered

in the modeling.

- Case Study D is a comprehensive field study of a cut slope in an expansive soil in

Zao-Yang, Hubie, China, which was instrumented by Ng et al. (2003) to investigate

the performance of the expansive soil slope over a period of one month. The effects

of soil cracks, daily climate data, along with two artificial rainfall events are

investigated.

- Case Study E is a field experiment in Arlington, Texas, USA, conducted by Briaud

et al. (2003) for measuring the shrink and swell movements of four full-scale spread

footings over a period of 2 years.

4.5 Summary

The modulus of elasticity based method (MEBM) is proposed for predicting the vertical

movements of unsaturated expansive soils with respect to time using the modulus of

elasticity as a tool. In summary, the proposed MEBM is a three step-procedure as shown

in Figure (4.3). The first step is to determine the matric suction changes for each time

PROPOSED APPROACH FOR PREDICTING VERTICAL MOVEMENTS OF EXPANSIVE SOILS 123

increment. The second step is to determine the modulus of elasticity that corresponds to

the matric suction value. The last step is to estimate the variation of the vertical soil

movements with respect to time.

To illustrate the step-by-step procedure of the MEBM and its validation for the prediction

of the vertical movements of expansive soils over time, the details of simulation of the

case studies under consideration are presented in the next chapter.

Fig. 4.3. Three step-procedure of the MEBM

Matric suction changes over time

Elastic modulus of unsaturated expansive

i

Vertical movements of expansive soils

CHAPTER 4 124

CHAPTER 5

VALIDATION OF THE PROPOSED MODULUS OF

ELASTICITY BASED METHOD

5.1 Introduction

The Modulus of Elasticity Based Method (MEBM) is proposed in this research study for

the prediction of the volume change movement of expansive soils over time. The MEBM

is tested in this chapter for its validity in several case studies collected and gathered from

the literature. The vertical soil movements associated with the volume change of

expansive soils for each case study are predicted using the MEBM as presented and

detailed in the preceding chapter. The pore air pressure is assumed to be equal to the

atmospheric pressure for all situations. Therefore, the net normal stress is equivalent to

the net total stress and the matric suction is equivalent to the absolute value of the

negative pore water pressure.

The following activities are carried out for each of the investigated case studies:

- Simulation of the matric suction variations over time using VADOSE/W program.

The program models the water flow through unsaturated soils and the soil-

atmospheric interactions over time, and then estimates the corresponding changes

in soil matric suction within the active zone depth.

- Estimation of the modulus of elasticity of unsaturated expansive soils using

Vanapalli and Oh (2010) model (i.e., VO model). The VO model was described

and validated in Chapter Three.

- Prediction of the vertical soil movements over time using the simplified volume

change constitutive relationship for soil structure (Equation 4.7). This is the first

125

time that a simplified constitutive relationship for the volume change of soil

structure has been used to estimate the soil movements in terms of the matric

suction changes and the corresponding values of the modulus of elasticity.

The details of soil movement calculations using the MEBM approach, and the analysis

and the discussion of the results of the MEBM for the case studies under consideration

are presented in this chapter.

5.2 Case Studies

The five case studies investigated in this chapter are summarized in Table (5.1), and

labelled as Case Study A, Case Study B, Case Study C, Case Study D, and Case Study E

for simplicity. These case studies are used to check the validity of the MEBM for the

estimation of the vertical soil movements (i.e., volume change behavior) of natural

expansive soils with respect time. The case studies are chosen to be representative of a

variety of site conditions from different regions of the world that include Canada, China,

and the United States. Several scenarios with different boundary conditions are used to

simulate each of these case studies. The predicted vertical movements of the five case

studies using the proposed MEBM are compared to the measured/estimated results that

are published in the literature.

Table 5.1. Case studies simulated using the proposed MEBM

Case study/ Reference Description Period of simulation

Case Study A (Vu and Fredlund 2006)

A problem example of a slab-on-ground placed on Regina expansive clay subjected to a constant infiltration rate.

175 days

Case Study B (Yoshida et al. 1983, Vu and Fredlund 2004 )

A light industrial building in north-central Regina, Saskatchewan, Canada, subjected to a leakage from a water line below a floor slab.

150 days

Case Study C (Ito and Hu 2011)

A test site in Regina, Saskatchewan, Canada. Various factors influencing the soil

One year

CHAPTER 5 126

movements (such as climate changes, vegetation, watering of lawn, and structural impact) have been considered in the simulation.

Case Study D (Ng et al. 2003)

A cut-slope in an expansive soil in Zao-Yang, Hubie, China, subjected to daily climate data with two artificial rainfall events. The effects of soil cracks and environmental conditions on the soil movements have been investigated.

One month

Case Study E (Briaud et al. 2003)

A field site in Arlington, Texas, USA, with four full-scale spread footings. Factors that influence the shrink and swell movements of expansive soils such as the daily weather, and the vegetation have been considered for this field construction site.

Two years

5.3 Case Study A: a Slab-on-Ground Placed on Regina Expansive Clay

Subjected to a Constant Infiltration Rate (Vu and Fredlund 2006)

Case Study A is an example problem of volume change of unsaturated expansive soils

used to validate the MEBM. Case Study A was originally modeled by Vu and Fredlund

(2006), considering 5 m thick deposit of Regina expansive clay that was partially covered

with a slab (i.e., lightly loaded structure). An infiltration of 2 × 10-8 m/s was imposed at

the ground surface around the structure over a period of 175 days (Figure 5.1). The soil-

water characteristic curve (SWCC) for Regina clay was given by Vu (2002) that fits the

experimental data measured by Shuai (1996). The soil permeability function (k function)

is estimated in this study using the software VADOSE/W, based on the input information

of the saturated coefficient of permeability (0.00523 m/day = 6.053 × 10-8 m/s) and the

SWCC given by Vu (2002). Figure (5.2) shows the SWCC and the permeability function

used for Case Study A. The Regina expansive clay properties are presented in Table

(5.2).

The heave estimation of Case Study A was performed in three steps as shown in Figure

(4.3). First, the changes in soil matric suction arising from the infiltration alongside the VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 127

slab were simulated using VADOSE/W. Second, the values of the unsaturated modulus

of elasticity associated with the changes in matric suction were estimated using the VO

model. Finally, the response of the unsaturated expansive soil during the infiltration (i.e.,

soil heave) was calculated from the simplified volume change constitutive equation for

soil structure (Equation 4.7).

Fig. 5.1. Geometry and boundary conditions of Case Study A (modified after Vu and Fredlund 2006)

Fig. 5.2. Hydraulic characteristics of Regina expansive clay used for Case Study A (SWCC data obtained from Vu 2002)

CHAPTER 5 128

Table 5.2. Mechanical properties of Regina expansive clay for Case Study A

(modified after Shuai 1996)

Soil properties Values Atterberg limits wl = 69.9%, wp = 31.9%, Ip = 38% Unified Soil Classification System CH, Inorganic clay of high plasticity Specific gravity Gs = 2.83 Maximum dry unit weight γdmax = 14.01 kN/m3 Optimum water content woptm = 28.5% Swelling index Cs = 0.088 Corrected swelling pressure Ps = 300 kPa

5.3.1 Simulation of matric suction changes over time

The simulation of matric suction variations in Regina clay underlying the slab-on-ground

was implemented using VADOSE/W over a period of 175 days. Case Study A was

modeled as a 2-D problem considering the transient isothermal analysis (i.e., the

temperature in the soil was assumed to be constant, T = 10 oC). The infiltration of

2 × 10-8 m/s was imposed at the ground surface around the structure over 175 days.

Figure (5.1) shows the initial and the boundary conditions of the simulation. For

comparison purposes, the initial and the boundary conditions assumed by Vu and

Fredlund (2006) for modeling Case Study A were used in the present simulation. A

matric suction value of 400 kPa was applied along the bottom boundary during the

infiltration. This was achieved by specifying a pressure head of -40.787 m (i.e., -400 kPa

/ 9.807 kN/m3) at the bottom boundary. The initial matric suction in the soil mass was

assumed to be equal to 400 kPa.

Vu and Fredlund (2006) modeled the soil heaves at three points A, B, and C at depths of 0

m, 1.5 m, and 3.5 m, respectively, located to the right of the outer edge of the slab as

shown in Figure (5.1). The same points were also modeled in this study using the

MEBM. Figure (5.3) shows the changes of matric suction with respect to time at the

locations A, B, and C. Figure (5.4) shows the variations of matric suction profiles at the

right of the outer edge of the slab in response to the infiltration over the period of

simulation.

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 129

Fig. 5.3. Matric suction changes with time for the three locations A, B, and C

Fig. 5.4. Matric suction profiles for various elapsed times at the right of the outer edge of the slab

CHAPTER 5 130

5.3.2 Estimation of soil modulus of elasticity associated with matric suction

The soil modulus of elasticity E and Poisson’s ratio µ are the two soil property

parameters required for the prediction of soil heave (see Equation 4.7). The Poisson’s

ratio µ does not significantly change with matric suction for expansive soils as they

desaturate at a very slow rate. Vu and Fredlund (2006) calculated the value of Poisson’s

ratio µ in terms of the coefficient of earth pressure at rest 0K from the following relation

0

01K

Kµ =

+ (5.1)

Lytton (1994) presented typical values for the coefficient of earth pressure at rest 0K

which were back calculated based on field observations of soil heave and shrinkage. The

coefficient of earth pressure at rest of 0.67, suggested for wetting conditions when cracks

in the soil are essentially closed, was used for this case study (Case Study A). Thus, the

Poisson’s ratio µ = 0.4 was calculated using equation (5.1).

The modulus of elasticity of unsaturated soils is not constant but is a function of both the

mechanical stress and the matric suction. The VO model (Equation 3.1) was extended in

Chapter Three to estimate the soil modulus of elasticity for unsaturated expansive soils.

The information required for using the model includes the SWCC and the two fitting

parameters (i.e., α and β ) along with the saturated modulus of elasticity Esat. The value

of the fitting parameters β = 2 recommended in Chapter Three for expansive soils was

used for this case study. The fitting parameter α = 1/9 was assumed to provide reasonable

comparisons between the predicted heave using the MEBM and Vu and Fredlund (2006)

method.

The saturated modulus of elasticity Esat of Regina expansive clay for the present case

study can be calculated based on the constitutive surface for soil structure in terms of the

soil void ratio e as shown in equation (5.2) (Zhang 2004, Vu and Fredlund 2006).

03(1 2 ) (1 )( ( ))sat

a

eEe u

µσ

− +=

∂ ∂ − (5.2)

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 131

where ( )auσ − is net normal stress, and e0 is initial void ratio.

Vu and Fredlund (2006) proposed equation (5.3) to fit the void ratio constitutive surface

for unsaturated expansive soils with six fitting parameters a, b, c, d, f, and g as follows

1 ( ) ( )log1 ( ) ( )

a a w

a a w

c u d u ue a bf u g u u

σσ

+ − + −= + + − + −

(5.3)

where ( )a wu u− is matric suction, and a, b, c, d, f, and g are fitting parameters of the void

ratio constitutive surface for Regina expansive clay (Equation 5.3) which are shown in

Table (5.3).

Table 5.3. Fitting parameters of the void ratio constitutive surface for Regina

expansive clay (Vu and Fredlund 2006)

Fitting parameters of equation (5.3) a b c d f g 1.2492 -0.0979 4.8240 3.3330 0.0009 0.0012

When the soil is saturated, equation (5.3) can be written as a function of only the net

normal stress ( )auσ − as follows:

1 ( )log1 ( )

a

a

c ue a bf u

σσ

+ −= + + −

(5.4)

Zhang (2004) also proposed equation (5.5) to fit the relationship of void ratio versus net

normal stress for saturated soils.

11

1

1

log ( )1 exp ( )a

ae y u xb

σ= +− −

+ − (5.5)

where 1a , 1b , 1x , and 1y are fitting parameters. In this analysis, equation (5.5) has been

used to fit the void ratio constitutive relationship for Regina expansive clay proposed by

Vu and Fredlund (2006) (Equation 5.4). The fitting parameters of equation (5.5) 1a , 1b ,

CHAPTER 5 132

1x , and 1y were obtained to be equal to 0.5085, -1.3154, 1.1773, and 0.8128, respectively.

Figure (5.5) shows the oedometer test results conducted by Shuai (1996) for Regina

expansive clay along with the two fitting equations (Equations 5.4 and 5.5). The two

best-fit equations are coincident as shown in Figure (5.5). Either equation (5.4) or

equation (5.5) can be used to calculate the saturated modulus of elasticity of Regina

expansive clay.

Fig. 5.5. Oedometer test results for Regina expansive clay along with the best fit equations

The calculation of the saturated modulus of elasticity Esat using equation (5.2) requires

the derivative of the relationship of void ratio versus net normal stress (Equation 5.4 or

Equation 5.5). The derivative of equation (5.5) is as follows (Zhang 2004)

2

1 11

1 1

1

log ( ) log ( )1 exp ( ) exp ( )

(( ) ( ) ln10

a a

a a

u x u xab bde

d u u b

σ σ

σ σ

− − − − −

+ − − = −

− − (5.6)

Then, the saturated modulus of elasticity can be calculated by combining equations (5.2)

and (5.6). The average value of the saturated modulus of elasticity of Regina expansive

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 133

clay for Case Study A was calculated to be about 1100 kPa for the initial void ratio e0 =

0.955, and Poisson’s ratio µ = 0.4.

Vu and Fredlund (2004) suggested that the saturated modulus of elasticity of soil can also

be calculated directly from the volume change index with respect to net normal stress

(swelling index) Cs. The saturated modulus of elasticity Esat can be expressed as a

function of the swelling index Cs, the initial void ratio e0, and Poisson’s ratio µ , which

can be written for the K0 loading condition as follows

02.303(1 ) (1 2 ) (1 ) ( )(1 )

µ µ σµ

+ − += −

−sat as

eE uC

(5.7)

For Regina expansive clay having e0 = 0.955, Cs = 0.088, and µ = 0.4, equation (5.7) can

be written as

23.876( )σ= −sat aE u (5.8)

The average value of the saturated modulus of elasticity of Regina expansive clay for

Case Study A was also obtained from equation (5.8) to be 1100 kPa. Table (5.4)

summarizes the soil properties used for Case Study A.

Table 5.4. Soil properties for Case Study A (modified after Vu and Fredlund 2006)

Soil properties Values Total unit weight γt = 17.27 kN/m3 Initial void ratio e0 = 0.955 Swelling index Cs = 0.088 Poisson’s ratio µ = 0.4 Saturated modulus of elasticity Esat = 1100 kPa Saturated coefficient of permeability ksat = 0.00523 m/day Saturated volumetric water content θs = 0.5015 Initial matric suction (ua - uw)i = 400 kPa

CHAPTER 5 134

Equation (3.1) was then solved for each time increment over the simulation period of

Case Study A, for the matric suction ( )a wu u− , the degree of saturation S, the saturated

modulus of elasticity satE , and the fitting parameters β = 2 and α = 1/9, to calculate the

unsaturated modulus of elasticity unsatE over time.

( )1/101.3a w

unsat sata

u uE E S

Pβα

− = +

(3.1)

5.3.3 Prediction of soil heave over time

The water infiltration into an unsaturated expansive soil leads to a decrease in matric

suction, which contributes to soil volume change predominantly in the vertical direction

(1-D heave). For Case Study A, to evaluate the 1-D heave at any depth over 175 days, the

5 m depth of soil profile was divided into five equal layers of 1 m thickness. Based on the

estimated matric suction changes within the soil profile (see Section 5.3.1), the total

heave at any depth for a certain time was computed. The day to day changes of the matric

suction values estimated using VADOSE/W were substituted into the volume change

constitutive relationship for soil structure (Equation 4.6) to calculate the 1-D heave of

each layer ih∆ associated with the time increment (i.e., a day).

(1 )(1 2 ) ( )(1 )i i a w

i

h h u uE

+ µ − µ∆ = ∆ − − µ

(4.6)

where ( )a wu u∆ − is change in matric suction for each time increment, and ih the

thickness of the ith layer.

The heave of each soil layer at a given time was then calculated as the cumulated value

of the heaves of the soil layer for all days prior to that given time. The total heave at any

depth h∆ was obtained from the summation of the heave of n layers beneath (Equation

4.7).

1 1

(1 )(1 2 ) ( )(1 )

n n

i i a wi i i

h h h u uE= =

+ µ − µ∆ = ∆ = ∆ − − µ

∑ ∑ (4.7)

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 135

Figure (5.6) shows the comparison of soil heaves predicted using the proposed MEBM

with the numerical modeling results published in Vu and Fredlund (2006) at the three

locations A, B, and C shown in Figure (5.1).

Fig. 5.6. Comparison between the predicted heaves using the proposed MEBM and Vu and Fredlund (2006) method at three locations A, B, and C

5.3.4 Analysis and discussion

The matric suction variations were evaluated at the three locations (A, B, and C) for the

period of simulation (175 days) using VADSOE/W. Review of Figure (5.3) shows that

the initial matric suction of 400 kPa decreases with time. The matric suction at the ground

surface (at A) has a lower value compared to the other two locations (B and C). This

reflects the effect of the water infiltration which is accompanied by a reduction in matric

suctions. The infiltration influences are primarily in the upper soil layers near the ground

surface, which contribute to significant changes in matric suction. Figure (5.4) shows the

variations of matric suction profiles at the right of the outer edge of the slab in response

to the infiltration during the period of simulation. This figure highlights the effect of the

infiltration on the soil matric suction as discussed earlier.

Based on the estimated matric suction changes over time, the soil heave was calculated

with respect to time at the three locations A, B, and C. Figure (5.6) shows the comparison

between the values of the 1-D heaves predicted using the proposed MEBM and the

CHAPTER 5 136

numerical modeling results of Vu and Fredlund (2006) for Case Study A. The total heave

increased with a decrease in matric suction with most of the heave occurring in the first

60 days. The coefficient of determination between the results of the MEBM and Vu and

Fredlund (2006) method was relatively high (R2 = 0.97). However, the predicted heaves

using the MEBM are slightly higher than the numerical modeling results of Vu and

Fredlund (2006). The reason for this difference may be attributed to the proposed MEBM

procedure which considers the 1-D heave, whereas the numerical modeling results of Vu

and Fredlund (2006) represents the soil heave in 2-D.

5.4 Case Study B: a Light Industrial Building in North-Central Regina,

Saskatchewan, Canada (Yoshida et al. 1983, Vu and Fredlund 2004)

Case Study B, a light industrial building constructed on an expansive soil deposit in

Regina, Saskatchewan, Canada, has been also used in this study for testing the validity of

the MEBM for the prediction of soil heave over time. The building was constructed by

the Division of Building Research, National Research Council, in 1961 as a part of a

comprehensive field study program to investigate the problems associated with

construction in/on expansive soils in the Regina area of Saskatchewan. One year after

construction, heaving and cracking in a floor slab were noticed by the building owner.

The owner also noticed an unexpected loss of 35 m3 of water. This amount of water loss

was traced to a leak in a hot-water line beneath the floor slab (Yoshida et al. 1983). The

maximum heave observed on the slab was found to be 106 mm. Figure (5.7) shows the

geometry and the boundary conditions for Case Study B. A 2.3 m thick deposit of Regina

expansive clay was considered for the estimation of 1-D heave. This depth was

considered to be equivalent to the active zone depth beyond which there will be no

tendency for swelling. It was assumed that water leaked from the pipe line along a 2 m

length (Yoshida et al. 1983) (see Figure 5.7). The matric suction was relatively high

close to the surface and decreased with depth. The matric suction through the soil profile

dissipated with time and reached steady state condition, in which the matric suction under

the center of the slab was equal to 20 kPa, in about 150 days (Yoshida et al. 1983).

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 137

The initial and the boundary conditions assumed by Vu and Fredlund (2004) for

modeling Case Study B are also used in the present simulation. Figure (5.8) presents the

SWCC and the coefficient of permeability function (k function) given by Vu and

Fredlund (2004). The SWCC was estimated using Fredlund and Xing (1994) equation

while the permeability function was obtained from Leong and Rahardjo (1997) equation.

The soil properties used for the case history analysis are summarized in Table (5.5). The

three steps of the MEBM, as stated previously, including the simulation of matric suction

changes over time, the estimation of unsaturated elasticity moduli associated with the

changes in matric suction, and the prediction of soil heave with respect to time, were

applied on Case Study B for various elapsed times (i.e., 5, 20, 50, 100 days, and at the

steady state condition).

Fig. 5.7. Geometry and boundary conditions of Case Study B (modified after Vu and Fredlund 2004)

CHAPTER 5 138

Fig.5.8. Hydraulic characteristics of unsaturated Regina expansive clay for Case Study B (modified after Vu and Fredlund 2004)

Table 5.5. Soil properties for Case Study B (Vu and Fredlund 2004)

Soil properties Values Atterberg limits wl = 77%, wp = 33%, Ip = 44% Specific gravity Gs = 2.82 Total unit weight γt = 18.88 kN/m3 Initial void ratio e0 = 0.962 Swelling index Cs = 0.09 Saturated coefficient of permeability ksat = 6.8×10-5 m/day Saturated volumetric water content θs = 0.493

5.4.1 Simulation of matric suction changes over time

The simulation of matric suction changes over time for Case Study B was carried out

using the VADOSE/W program. The case study was simulated as a 2-D problem

considering the transient isothermal analysis to solve the system of equations for the

unsaturated flow. The boundary conditions for the matric suction simulation are

presented in Figure (5.7). The leak along the 2 m length of the hot-water line beneath the

floor slab was represented by specifying zero pressure head at the desired line nodes. A

“no flow” natural boundary condition on the floor slab was applied by default in

VADOSE/W to represent the slab as an impervious layer. A matric suction of 12 kPa

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 139

was applied along the bottom boundary during the period of simulation (150 days), which

was achieved by specifying a pressure head of -1.223 m (i.e., -12/9.807) at the bottom

boundary. A matric suction of 888 kPa (i.e., -90.52 m pressure head) was applied along

the top boundary around the slab. The soil temperature was assumed to be constant (T =

10 oC); in other words, the temperature effects were omitted from the simulation (i.e.,

transient isothermal analysis). The initial matric suction given by Vu and Fredlund (2004)

as a function of depth was used to represent the initial condition of the analysis (see

Figure 5.7).

Vu and Fredlund (2004) predicted the changes of matric suction with time at 1.3 m depth

at three locations D, E, and F shown in Figure (5.7). The changes of matric suction for

the same locations were also obtained from VADSOE/W. Figure (5.9) shows the

comparison of the predicted matric suction values over time at D, E, and F using

VADSOE/W with those published in Vu and Fredlund (2004). Figure (5.10) presents the

comparison of the predicted matric suction profiles under the center of the slab for

various elapsed times (i.e., 5, 20, 50, 100 days, and at the steady state condition).

Fig. 5.9. Matric suction changes with time for the three locations D, E, and F

CHAPTER 5 140

Fig. 5.10. Matric suction profiles for various elapsed times under the center of the slab

5.4.2 Estimation of soil modulus of elasticity associated with matric suction

The soil modulus of elasticity associated with the day to day changes of matric suction

was estimated using the VO model (Equation 3.1). The Poisson’s ratio µ of 0.4,

suggested by Vu and Fredlund (2006) for the coefficient of earth pressure at rest 0K of

0.667, was used in this analysis. The recommended value of the fitting parameter β = 2

was used for this case study. The fitting parameter α was assumed to be 1/13 in order

to provide a reasonable comparison between the predicted and the measured soil

heaves. The saturated modulus of elasticity Esat was calculated based on the analysis of

the oedometer tests results for Regina expansive clay as discussed in section (5.3.2).

Equation (5.2) was combined with the best-fit equation of the void ratio constitutive

relationship for the saturated Regina expansive clay (Equation 5.4 or Equation 5.5) to

calculate the saturated modulus of elasticity Esat. In addition, equation (5.7) was also used

to calculate the saturated modulus of elasticity in terms of the swelling index Cs. The

average value of the saturated modulus of elasticity of Regina expansive clay for Case

Study B was calculated to be equal to 550 kPa. The variations of the unsaturated

modulus of elasticity Eunsat with respect to matric suction over the period of simulation

was estimated using equation (3.1).

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 141

5.4.3 Prediction of soil heave over time

To evaluate the soil heave resulting from the leakage in the water line below the floor

slab, the 2.3 m depth of the soil (depth of matric suction variations) was subdivided into

six equal layers of 0.3 m thickness and a bottom layer of 0.5 m thickness. The day to day

changes of the matric suction value estimated using the VADOSE/W program were

substituted into the volume change constitutive relationship (Equation 4.6) to calculate

the soil layer heave for each day. The total heave at a certain depth for a given time was

computed by adding the heave associated with that time for all layers below the

considered depth (Equation 4.7). Figures (5.11) and (5.12) show the predicted soil heaves

using the MEBM under the center of the slab and along the surface of the slab,

respectively, compared with both the measurements and the numerical modeling results

published in Vu and Fredlund (2004).

Fig. 5.11. Predicted and measured soil heave profiles under the center of the slab

CHAPTER 5 142

Fig. 5.12. Predicted and measured soil heave values along the surface of the slab

5.4.4 Analysis and discussion

Figure (5.9) shows the matric suction variations with respect to time at the three locations

D, E, and F. It can be seen from Figure (5.9) that the initial matric suction significantly

decreases with time and approaches a steady state condition in about 150 days. A

reasonable agreement was observed between the matric suction values obtained from

VADOSE/W and estimated by Vu and Fredlund (2004). The coefficient of determination

was relatively high (R2 > 0.89). However, the predicted matric suction at E (below the

leakage position) obtained from the VADSOE/W program is slightly higher compared to

the matric suction values predicted by Vu and Fredlund (2004) at the same location. The

reason for this difference may be attributed to the use of different software to simulate the

matric suction variations. VADOSE/W was used in the present study while FlexPDE2

(the general-purpose partial differential equation solver) was used by Vu and Fredlund

(2004). Also, the differences may be due to the effect of the boundary condition that has

been used to represent the leakage (i.e., uw = 0). Review of Figure (5.9) also shows that

the matric suction at E has a lower value compared to the other two locations D and F.

This reflects the effect of wetting which is accompanied by a reduction in matric suction.

Figure (5.10) shows the variations of matric suction profiles at the center of the slab in VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 143

response to the wetting over the period of simulation. The results of VADOSE/W have a

good match with the estimations of the matric suction variations from Vu and Fredlund

(2004) method (R2 > 0.97). The wetting primarily influences in the upper soil layers near

the ground surface, which contributes to significant changes in matric suction.

Figure (5.11) shows a close agreement between the soil heave under the centre of the slab

estimated using the proposed MEBM and Vu and Fredlund (2004) method with high

coefficient of determination (R2 > 0.98). The total heave increases with a decrease in

matric suction, with most of the heave occurring in the upper soil layers near the ground

surface where the changes of soil suction are relatively high. The heaves measured by

Yoshida et al. (1983) have been also plotted on the same graph (Figure 5.11). Some

differences between the predicted and the measured heave can be observed. The heave

values measured at depths of 0.58 and 0.85 m correspond to the predicted heave at 100

days. Figure (5.12) compares the predicted soil heaves from the MEBM, the predicted

soil heaves from Vu and Fredlund (2004) method, and the measured total heave at the

surface of the slab. The soil heave predicted using the MEBM agrees well with the

published data (measurements/estimates) (R2 > 0.94).

5.5 Case Study C: a Test Site in Regina, Saskatchewan, Canada (Ito and

Hu 2011)

The validity of the MEBM for predicting the vertical soil movement (i.e., heave/shrink)

with respect to time is tested in an additional test site (Case Study C), taking account of

the soil-environment interactions. The test site of Case Study C was previously modeled

by Ito and Hu (2011) as a part of a program of studying the performance of asbestos

cement (AC) water mains in Regina, Saskatchewan, Canada. Ito and Hu (2011) predicted

the vertical movements of the site soil based on the matric suction data predicted from a

soil-atmosphere coupled model by applying one year’s climate data (1 May, 2009−30

April, 2010). Various factors that influence the soil movements such as climate,

vegetation, watering of lawn, and soil cover type were considered in modeling the case

study. However, Ito and Hu (2011) approach requires oedometer tests which are costly

CHAPTER 5 144

and tedious for determining the elasticity parameters functions required for the soil–

displacement analysis.

Case Study C was modelled in this section using the MEBM approach which, as

previously illustrated, integrates the simplified constitutive relationship for soil structure

along with the soil-atmospheric interactions model to estimate the vertical movements

with respect to time. The estimated values of matric suction, volumetric water content,

and vertical movement of expansive soils at different depths obtained from the MEBM

approach were compared with the published results of Ito and Hu (2011). The description

of the investigated test site and the details of its modeling analysis using the proposed

MEBM approach are presented below.

5.5.1 Site description

The test site is located in a residential area with a high water main breakage rate in the

city of Regina, Saskatchewan, Canada. It includes a park area with thick grass of 100 mm

and a wide paved road with 150 mm thick asphalt pavement (Figure 5.13). The

stratigraphy of the site consists of 6.4 m of highly plastic clay (wl varies from 70 to 94%

with Ip of 40 to 65%), 1.8 m of silt, and 6.8 m of till as shown in Figure (5.14). The

choice of thickness and soil properties for each layer was guided by field observations

from Vu et al. (2007). Figures (5.15) and (5.16) show the soil-water characteristic curve

(SWCC) and the permeability function (k function) for each soil (Ito and Hu 2011). The

properties for each soil used in modeling the soil-environment interactions for this case

study are summarized in Table (5.6).

Table 5.6. Soil properties used for Case Study C (Ito and Hu 2011)

Soil properties Clay Silt Till Dry unit weight, γd (kN/m3) 12.0 13.8 15.1 Initial void ratio, e0 1.2 0.9 0.7 Saturated coefficient of permeability, ksat (m/s) 9 × 10-9 7.56 × 10-6 10 × 10-10 Saturated volumetric water content, θs 0.56 0.48 0.42 Swelling index, Cs 0.09 0.09 0.09 Poisson’s ratio, µ 0.33 0.3 0.3

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 145

Fig. 5.13. Schematic of the test site of Case Study C (modified after Ito and Hu 2011)

Fig. 5.14. Soil profile and soil properties of Case Study C (modified after Ito and Hu 2011)

CHAPTER 5 146

Fig. 5.15. Soil-water characteristic curves of the site soils for Case Study C (modified after Ito and Hu 2011)

Fig. 5.16. Permeability functions of the site soils for Case Study C (modified after Ito and Hu 2011)

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 147

Figure (5.17) shows the climate data over a period of one year from 1 May, 2009 to 30

April, 2010 obtained from a weather station at Regina international airport, located at 5

km from the site (Ito and Hu 2011). The measured daily precipitation for the investigated

area over the study period shows that the majority of storm events occurred during the

summer, while in winter season (1 November, 2009 to 31 March, 2010) the precipitation

was received as snow. The recorded average daily temperature varied from -10 °C (from

November to March) to 10 °C (from April to October). The maximum difference between

the high temperature (26 °C) and the low temperature (-28 °C) was 54 °C. The wind

speed had no clear trend and varied between 2 and 14 m/s with an average value of 5 m/s.

The relative humidity data illustrated a high average of 78% during winter and a low

average of 64% during summer. The net radiation data was estimated by Ito and Hu

(2011) from the corrected solar radiation; the average net radiation was 6.4 MJ/m2/day in

winter and 13.8 MJ/m2/day in summer. This climate data was applied at the vegetative

cover as climate boundary over the period of simulation (from 1 May, 2009 to 30 April,

2010).

CHAPTER 5 148

Fig. 5.17. Climate data for the Regina test site of Case Study C (modified after Ito and Hu 2011)

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 149

5.5.2 Simulation of matric suction changes over time

To simulate the soil-atmospheric interactions and estimate the corresponding matric

suction changes, the soil profile shown in Figure (5.14) was modelled using the fully

coupled transient analysis with the VADOSE/W software. Beside the soil properties, the

initial and boundary conditions were required as input data. The initial conditions for all

nodes of the model domain, including the soil matric suction and the soil temperature,

were derived from implementing a steady-state analysis using the same model. Based on

the field data measured by Vu et al. (2007), the initial matric suction during the steady-

state analysis was set up to be 1600 kPa for the top 3 m of the clay layer, 1000 kPa for the

rest of the clay, 600 kPa for the silt, and 2000 kPa for the till. These values of matric

suction were achieved by specifying a pressure head of -163.15, -101.97, -61.18, and -

203.94 m, respectively. The soil temperatures of nodes at the lower boundary were set up

to be 10 °C.

For the fully coupled transient analysis, the climate and the vegetation data of the site

were applied on the vegetated area. A “no flow” natural boundary condition was applied

by default in VADOSE/W to represent the pavement as an impervious layer, while in and

out moisture flows are occurring through the vegetated area. The climate data shown in

Figure (5.17) was used for the simulation. However, the climate data in winter was set up

to be basically constant; the temperature was assumed to be -5 °C, the relative humidity

as 100%, and the other components of the climate data were set up to be zero. The

cumulated winter snow precipitation was applied on a single day (1 April, 2010), when

the temperature rose and remained above 0 °C. In other words, the model was not

intended to simulate the soil movement activities during winter. In addition, since the site

is located in a residential area with a park that has mature trees, Vu et al. (2007) and Ito

and Hu (2011) specified the site vegetation as good grass and reduced the daily wind

speed, precipitation, and net radiation recorded at the weather station by the scale factors

of 0.3, 0.7, and 0.3, respectively. Furthermore, a park watering rate of 1.8064× 10-3 m/day

was applied every Monday and Friday for the period from 23 June to 12 October as

reported in Vu et al. (2007). However, the water uptake by mature trees was not included

in the simulation. The vegetation data for the site, as given by Ito and Hu (2011), includes

the growing season starting in April and ending in October, the leaf area index function

CHAPTER 5 150

(LAI) for good vegetation with a maximum LAI value of 2 as given in SoilCover

(Unsaturated Soils Group 1996), the root depth that was suggested to be 150 mm with a

triangular root distribution, and the plant moisture limiting point and the wilting point

that were assumed to be 500 kPa and 2500 kPa, respectively.

The mass balance checking was performed on the VADOSE/W runs, and the model was

solved with a total mass balance error of less than 1.5%. The responses of both the soil

matric suction and the volumetric water content within the active zone to a changing

surface boundary over the entire year were predicted. Figures (5.18) and (5.19) show the

responses of the matric suction and the volumetric water content, respectively, under the

centre of the vegetation cover. The values predicted using VADOSE/W were compared

with the results of Ito and Hu (2011) as shown in Figures (5.18) and (5.19). Figure (5.20)

presents the corresponding matric suction profiles for different times under the centre of

the vegetation cover.

Fig. 5.18. Soil matric suction changes with respect to time at different depths under the centre point of the vegetation cover for Case Study C

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 151

Fig. 5.19. Volumetric water content changes with respect to time at different depths under the centre point of the vegetation cover for Case Study C

Fig. 5.20. Predicted matric suction profiles using VADOSE/W at different times under the centre point of the vegetation cover for Case Study C

CHAPTER 5 152

5.5.3 Estimation of soil modulus of elasticity associated with mtric suction

The modulus of elasticity associated with the matric suction changes, required for

predicting the soil movements, was estimated using the VO model (Equation 3.1).

Poisson’s ratio µ = 0.33 was used for the clay layer of the site as suggested by Ito and

Hu (2011). The fitting parameters β = 2 and α = 1/14.5 were chosen in order to provide

a reasonable comparison between the predicted and the published results of the vertical

movements of the clay layer as shown in the next section. The saturated modulus of

elasticity Esat for the clay layer was calculated directly from the volume change index

with respect to the net normal stress (i.e., swelling index Cs) using equation (5.7). For e0

= 1.2, Cs = 0.09, and µ = 0.33, equation (5.7) can be written as

38( )σ= −sat aE u (5.9)

The average value of the saturated modulus of elasticity for the top 4.3 m of the clay

layer was calculated to be 1000 kPa, and that for the remainder of the clay layer was 1600

kPa. Equation (3.1) was then solved based on the matric suction changes, the SWCC in

terms of the degree of saturation, and the average values of the saturated modulus of

elasticity to calculate the unsaturated modulus of elasticity Eunsat associated with the

matric suction value. The estimated values of the unsaturated modulus of elasticity

provide a reasonable comparison between the predicted and the published results of the

vertical soil movements for the investigated site as presented in the following section.

5.5.4 Prediction of vertical soil movement over time

Once the soil suction changes and the associated modulus of elasticity with respect to

time are estimated, the vertical soil movements can be calculated at any depth and time.

The Regina expansive clay layer of Case Study C was divided into 14 sub-layers (the top

2 sub-layers with 0.05 m thickness, and the other 12 sub-layers with 0.5 m thickness).

The thickness of the sub-layers was chosen such that the investigated locations by Ito and

Hu (2011) (i.e., 0, 0.5, 1, 2, 3, and 6 m depth) were located at the middle of the suggested

soil sub-layers.

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 153

The vertical soil movement at a certain depth for each day was computed by adding

the daily vertical soil movement for all layers below the considered depth. Figure (5.21)

shows the vertical soil movement for each day at different depths below the centre of

the vegetated cover. The vertical movement of each layer at a given time was calculated

as the cumulated value of the soil layer movements for all days prior to that given time.

The total vertical movement at any depth was obtained from the summation of the

vertical movement of n layers beneath (Equation 4.7). Figure (5.22) presents the

variations of the total vertical soil movement over time at different depths below the

centre of the vegetation cover.

Fig. 5.21. Predicted vertical movements of clay layer for each day at different depths below the centre of the vegetation cover using the MEBM

CHAPTER 5 154

Fig. 5.22. Predicted total accumulated vertical soil movements at different depths under the centre of the vegetation cover using the MEBM

5.5.5 Analysis and discussion

Figures (5.18) and (5.19) summarize the predicted matric suction and the volumetric

water content, respectively, under the centre of the vegetation cover in response to the

changes in the surface boundary. These predicted values were found to vary with

depth and time, and correlated well with the environmental condition on the surface

boundary. The matric suction and the corresponding volumetric water content at the

ground surface fluctuated widely and these fluctuations reduced with depth. There

was a reasonable agreement between the predicted values of matric suction and

volumetric water content of this study and Ito and Hu (2011) study (the determination

coefficient R2 > 0.75). The correspondence between the results was accomplished

using the same meteorological data (i.e., precipitation, temperature, etc.), soil

properties, and initial and boundary conditions.

Figure (5.20) shows the matric suction profile at various times under the centre of the

vegetation cover of the investigated site, which represents the key information of the

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 155

MEBM approach. Extreme changes in the matric suction (vary between 600 and 2500

kPa) occurred at the ground surface. During evaporation (in summer season), the

matric suction has relatively greater values at the surface, and the remainder of the

profile adjusts accordingly. During infiltration (in the spring), the matric suction

decreases at the surface, and it continues to decrease as water infiltrates to greater

depths. The soil suction fluctuations were predominant at the surface and diminished

at about 3.4 m. According to Azam and Ito (2012), such a behavior may be attributed

to the interaction of environmental factors on the clay layer. The surface layer at an

initially unsaturated state readily imbibes any water made available by the infiltration.

Likewise the surface layer can rapidly lose water under the evaporation. With

increasing depth, the overlaying soil provides a cover and the geotechnical properties

of the underlying materials become progressively more significant. The high water

retention capability and the low coefficient of permeability of the clay, especially

under unsaturated conditions, impede variations in the suction at higher depths. The

modeling results of the soil-environmental interaction in this study corroborated well

with the results of Ito and Hu (2011) model thereby validating the VADOSE/W

simulation. Overall, the top 3.4 m depth of this soil profile was found to be the active

zone in which the environmental factors (infiltration and evaporation) are

predominant.

The soil movement was predicted as a function of time and depth. Both upward and

downward soil movements (i.e., heave and shrinkage) were observed over the period

of the study. The soil movement has a strong correlation with the predicted values of

matric suction. The soil swells with a decrease in matric suction and shrinks with an

increase in matric suction. The soil movements (heave/shrink) are predominant in the

upper soil layers near the ground surface in which the matric suction changes are

relatively high. In addition, the vertical soil movement clearly responds to the

climatic trend (i.e., infiltration and evaporation events). Figure (5.21) shows the

vertical soil movement for each day at different depths under the centre point of the

vegetated cover along with the daily precipitation over the period of simulation. The

fluctuation in the daily vertical soil movement is active because of the higher rate of

storm events. However, in winter (1 November, 2009 to 31 March, 2010), the soil

CHAPTER 5 156

response exhibited negligible variations as the precipitation was received as snow

(that piled up on the ground and did not infiltrate). Figure (5.23) compares the vertical

soil movements for each day at the ground surface and the 0.5 m depth estimated

using the MEBM approach with those estimated using Ito and Hu (2011) model. The

agreement between the results of two methods was reasonable (R2 > 0.88). Some

observed differences may be attributed to using different governing equations to

estimate the soil suction profiles and the corresponding soil movements.

Fig. 5.23. Comparison of the vertical soil movement for each day at the ground surface and the 0.5 m depth predicted using the MEBM and Ito and Hu (2011) model

Figure (5.22) summarizes the total vertical soil movements at different depths below

the centre point of the vegetation cover. The fluctuations in the total vertical soil

movements were relatively large near the ground surface; however, it ceased

completely at a depth of 6 m from the ground surface. The maximum heave

(maximum upward soil movement) and the maximum shrinkage (maximum

downward soil movement) at the ground surface beneath the center of vegetative

cover were predicted to be 10.2 mm and −3.6 mm, respectively. The total vertical soil

movement was estimated as 13.8 mm, which is the difference between the maximum

values of soil heave and soil shrinkage. According to Ito and Hu (2011) results, the

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 157

maximum values of heave and shrinkage were 8 mm and −5 mm, respectively, and the

total vertical soil movement was 13 mm. It can be seen that the total soil movements

predicted using the proposed MEBM was close to that predicted by Ito and Hu (2011)

with a 6% difference.

5.6 Case Study D: a Cut-Slope in an Expansive Soil in Zao-Yang, Hubie,

China (Ng et al. 2003)

A comprehensive field study investigated by Ng et al. (2003) has been chosen to

demonstrate the validity of the MEBM for predicting the field measurements of soil

movements with respect to time. The field study is a classic case in which the effect of

climatic conditions, soil properties, and soil cracks are considered on the volume change

behavior of expansive soils over one month (13 August to 12 September, 2001). This

field study was originally performed by Ng et al. (2003) to investigate the complex soil-

water interaction associated with the rainfall infiltration into a cut-slope in an expansive

soil in Zao-Yang, Hubie, China (i.e., the interaction among the changes of pore water

pressure (i.e., matric suction), water content, and soil movement as a result of rainfall

infiltration). This field study was meant to assist in the engineering design of the 180 km

portion of a canal to be excavated in unsaturated expansive soils along the middle route

of the South-to-North Water Transfer Project (SNWTP).

5.6.1 Description of the field study

Figure (5.24) shows the 11 m high cut slope in expansive clay in Zao-Yang, Hubie, China

(semi-arid area). The deposit is medium plastic, unsaturated, expansive clay that exhibits

significant volume changes as water content changes. The initial void ratio of the soil site

e0 varies from 0.63 to 0.69, and the soil plasticity index Ip is 31%. The basic physical

properties of soil specimens taken from the research slope at a depth of 1.0 m are

summarized in Table (5.7). The upper soil layer with a thickness varying from 1.0 m to

1.5 m is rich in cracks and fissures likely related to the swelling and shrinkage

phenomenon associated with expansive soils. Figure (5.25) shows the distributions of

cracks and fissures observed within the exposed surface near the monitoring area. The

maximum depth and the maximum width of the open cracks were estimated to be

CHAPTER 5 158

approximately 1.2 m and 10 mm, respectively. Figure (5.26) shows the in-situ

relationship between the soil water content and the matric suction, along with the soil-

water characteristic curve (SWCC) for the soil of the slope. The hysteresis between the

desorption and the adsorption curves appears to be relatively insignificant since the soil

specimens had experienced many wetting/drying cycles in the field (Zhan et al. 2006).

The coefficient of permeability for the soil site is relatively low (less than 10-7 m/s) (Zhan

2003).

Fig. 5.24. Cross-section of the instrumented slope of Case Study D (modified after Ng et al. 2003)

Table 5.7. Physical properties of soil specimens taken from the research slope at a depth

of 1.0 m (data from Zhan et al. 2007)

Clay grain content, (%)

Specific gravity, Gs

Dry unit weight, γd (kN/m3)

Initial void ratio, e0

Liquid limit, wL(%)

39 2.67 14.72-15.90 0.712 50.5

Plasticity Index, Ip (%)

Saturated permeability,

ksat (m/s)

Compressibility index, Cc

Swelling index, Cs

Poisson’s ratio, μ

31 10-10 − 10-7 0.13 0.025 0.4

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 159

Fig. 5.25. Cracks and fissures in an excavation pit near the monitoring area of Case Study D (Zhan et al. 2007)

Fig. 5.26. Soil-water characteristic curves for the slope soil (modified after Zhan et al. 2006)

Two artificial rainfall events were created by Ng et al. (2003) during the one month

period of field investigation including monitoring the induced rainfall infiltration through

the cut-slope. Figure (5.27) shows the two simulated rainfall events during the period of

monitoring (13 August to 12 September, 2001) with an average daily rainfall of 62 mm.

The first rainfall lasted for 8 days from 18 to 25 August, 2001. The second simulated

CHAPTER 5 160

rainfall was applied from 8 to 10 September, 2001. To understand the soil-water

interaction in an unsaturated expansive soil slope subjected to rainfall infiltration, a

comprehensive instrumentation and monitoring program was carried out by Ng et al.

(2003). The instrumentation included jet-filled tensiometers, thermal conductivity suction

sensors, moisture probes, earth pressure cells, inclinometers, vertical movement points,

an artificial rainfall simulator, a tipping bucket rain gage, a V-notch flow meter, and an

evaporimeter. The outcome of the field monitoring included a collection of considerable

amount of valuable data involving pore water pressure (or matric suction), soil water

content, and soil movements (heave/shrink) as a result of rainfall infiltration. More details

with respect to the instrumentation are available in Ng et al. (2003) and Zhan (2003).

Fig. 5.27. Intensity of rainfall events during the monitoring period of Case Study D (modified after Ng et al. 2003)

The step-by-step procedure of the proposed MEBM (Figure 4.2) is applied in the

following sections on the research slope of Case Study D to predict the vertical soil

movement with respect to time.

5.6.2 Simulation of matric suction changes over time

The matric suction changes associated with the rainfall infiltration through the

research slope were simulated using the VADOSE/W program. The program

primarily requires the input of appropriate in-situ soil properties and boundary

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 161

conditions to model the unsaturated and saturated flow through the slope. The

research slope was modeled as a 2-D problem using the fully coupled transient

analysis. The initial condition of the model was established by conducting the steady-

state analysis. The ground water table in the field was located at about 6 m depth from

the ground surface. To simulate the initial condition of the ground water table at an

average depth of 6 m, the pressure head in the top 1.5 m soil layer near the ground

surface (i.e., the tension cracks zone) was set up to vary from -7 to -1 m while the

pressure head value of the bottom boundary was set up at 1 m. As the daily

fluctuation of atmosphere temperature was insignificant at the considered site, the soil

slope was modeled using the simplified isothermal model. In other words, the thermal

properties of the slope soil were assumed to be constant (i.e., the soil temperature =

27 oC, the thermal conductivity = 400 kJ/days/m/°C, and the volumetric heat capacity

= 1875 kJ/m³/°C). The effect of vegetation on the volume change behavior of

expansive soils was omitted in the analysis since the top soil to a depth of about 10

mm of the slope was removed. The two artificial rainfall events with the daily climate

data obtained from a weather station in Wuhan city were applied as a climate

boundary at the surface layer of the slope. Figure (5.28) shows the 2-D model for the

research slope.

Fig. 5.28. 2-D model for the research slope of Case Study D

Distance (m)

Elev

atio

n (m

)

15

0

10

5

0

10 -5

20 30 40

CHAPTER 5 162

5.6.2.1 Model calibration

For Case Study D, VADOSE/W was calibrated to simulate the saturated and

unsaturated flow through the research slope. Valid assumptions with respect to the

soil properties and the boundary conditions were made such that reasonable

comparisons were achieved between the predicted and measured values of the water

flow properties. Similar approaches were used by Overton et al. (2006) and Diewald

(2003) for the model calibration. The upper portion of the slope with a thickness

varying from 1.0 to 1.5 m was observed to have cracks and fissures. Figure (5.26)

shows the SWCC of the slope soil in terms of the volumetric water content that was

calculated based on the SWCC in terms of the gravimetric water content estimated

using the best-fit Fredlund and Xing (1994) equation. Since the abundance of cracks

and fissures influences the SWCC behavior of expansive soils, a unique SWCC is not

reasonable to represent the expansive soil having tension cracks. For this modeling

task, the site soil was modeled by subdividing the tension cracks zone of the slope

into 3 layers as shown in Figure (5.29a). VADOSE/W was calibrated by varying the

SWCCs and the associated coefficient of permeability functions for each layer until

the estimated values of pore water pressure (PWP) matched the measurements.

Different scenarios using different assumptions related to the number of soil layers

and the soil properties were implemented until the behavior of the research slope was

successfully simulated. Figures (5.29b and c) show the SWCCs and the coefficient of

permeability functions of the soil layers obtained from the model calibration

including the cracking effects (bi-model) (i.e., two air-entry values). Figure (5.30)

shows the predicted and the measured pore water pressure versus time at the mid-

slope (R2) in response to the simulated rainfalls. The close agreement between the

predicted and measured values of the pore water pressure at the mid-slope (R2)

provides credence to the simulation based on the assumptions used in this analysis

(the coefficient of determination R2 = 0.74). The model can hence be considered to be

calibrated.

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 163

(a) 4-soil layers considered

(b) Soil-water characteristic curves of the soil layers of the slope

Layer (1) (0.4 m)

Layer (2) (0.9 m)

Layer (3) (0.2 m)

Layer (4) (5.6 - 13 m)

Tension cracks

zone (1.5 m)

CHAPTER 5 164

(c) Coefficient of permeability functions of the soil layers of the slope

Fig. 5.29. Key soil properties of the 4-layers considered for the research slope: (a) 4-soil layers considered, (b) soil-water characteristic curves, and (c) coefficient of permeability functions of the slope soil

Fig. 5.30. Comparison of the predicted and the measured pore water pressure (PWP) during the rainfall events at different depths at the mid-slope

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 165

Review of Figure (5.30) shows that both the predicted and the measured pore water

pressure within the crack tension zone were negative prior to the first artificial

rainfall. As expected, the negative pore water pressures near the ground surface were

higher than those at greater depth. After the application of the first rainfall event, the

predicted pore water pressures were generally consistent with the measurements; both

the predicted and the measured values of the pore water pressure changed from

negative to positive. The positive pore water pressure appeared in the slope soil. After

the end of the first rainfall, a recovery of negative pore water pressure was observed

in the predicted values as the measured values. The recovered negative pore water

pressures were much lower than the corresponding values prior to the application of

the rainfall. At the application of the second rainfall, the pore water pressure values

were similar to those at the first rainfall event. The predicted pore water pressure

reached the equilibrium condition faster than the measured values. This can be

attributed to the delay in the measured responses of the pore water pressure using the

instrumentation which required at least one and a half days’ time after the induced

rainfall infiltration. This delay was attributed by Ng et al. (2003) to the effect of the

abundant cracks and fissures in the soil on the soil-water interactions. Cracks

provided an easy pathway for rainwater to infiltrate the soil. The rainwater first

bypassed the regions between cracks as it entered directly through cracks. While

rainwater initially flowed through cracks and fissures, the tensiometers did not

register any significant changes of matric suction, and this continued until the

infiltrated rainwater filled the cracks (Ng et al. 2003). In contrast, the predicted pore

water pressure response was instantaneous after the commencement of the rainfall. In

spite of the discussed limitations that are difficult to be introduced into the modeling,

reasonable comparisons were observed between the predicted and the measured

values of the pore water pressure (the determination coefficient R2 = 0.74).

5.6.2.2 Model validation

The data from the field investigation performed by Ng et al. (2003) can be used for

the validation of the model. The model validation involves the comparisons of the

predicted and the measured properties of unsaturated and saturated flow through the

soil, including temperature, evaporation, matric suction, and volumetric water

CHAPTER 5 166

content. A good comparison between the measured and predicted properties

demonstrates that the calibrated model is reliable and can be used for predicting the

water flow and soil volume change behavior considering other scenarios. The

validation in the present analysis was performed by comparing the predicted and the

measured values of volumetric water content (hereafter referred to as VWC) for the

slope soil at section R2. Figure (5.31) shows the variations of the predicted and the

measured VWC with time at R2 in response to the simulated rainfalls. The response

of the VWC values was generally consistent with the corresponding pore water

pressures. Prior to the first rainfall, the predicted and the measured VWCs increased

with depth due to the influence of evaporation. After the rainfall, the predicted VWCs

simultaneously reached the equilibrium condition; however, the measured values have

the same delayed response as previously shown for the measured pore water

pressures. After the cessation of the first rainfall, the predicted VWCs began to

decrease progressively towards new equilibrium values, while the measured VWCs

remained constant before progressively reaching the equilibrium condition. At the

application of the second rainfall, the VWCs were similar to those at the first rainfall

event. In general, there was a reasonable agreement between the predicted and

measured values of VWC (R2 = 0.78). However, slight differences may be attributed

to the difficulty of the modeling of the natural variations in soil nature, composition,

and cracks. The inconsistency in part may also be related to the accuracy of the VWC

measurements in the field, which can be significantly affected by the soil compaction,

soil density and soil cracks as discussed in Ng et al. (2003) and Zhan (2003).

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 167

Fig. 5.31. Comparison of the predicted and the measured VWC during the rainfall events at different depths at the mid-slope

5.6.2.3 Simulation of matric suction fluctuation

After both the calibration and the validation of the model using VADOSE/W were

successfully achieved, the matric suction profiles were estimated for the period of

simulation (i.e., 30 days) at different sections of the slope. Figure (5.32) shows the

fluctuation of matric suction profiles at the middle section of the slope (R2) in

response to the rainfalls through the period of 30 days. The climate condition

primarily influenced the top 2.5 m near the ground surface and resulted in the

fluctuation of the soil matric suction within the active zone depth (i.e., the depth of

the active zone for this site is about 2.5 m). Review of Figure (5.32) shows that the

value of predicted matric suction decreased significantly after the commencement of

the rainfall due to an increase in the positive pore water pressure. The continued

rainfall resulted in further decrease in the matric suction till it disappeared. The

largest positive pore water pressure was observed at approximately 1.5 m depth. This

indicates the presence of a perched ground water table at that depth as observed in the

field measurements conducted by Ng et al. (2003).

CHAPTER 5 168

Fig. 5.32. Matric suction profiles at the middle of the slope for Case Study D

5.6.3 Estimation of soil modulus of elasticity associated with matric suction

The VO model (Equation 3.1) was used for estimating the modulus of elasticity

associated with the changes in matric suction. Poisson’s ratio µ for the slope soil was

assumed to be a constant value of 0.4. The fitting parameters β = 2 and α = 1/20 were

assumed to predict the vertical soil movements of the slope. The saturated modulus of

elasticity Esat for the slope soil was calculated using equation (3.8). Equation (3.8), as

described in Chapter Three, was developed based on the results of the triaxial tests

conducted by Zhan (2003) on saturated compacted specimens of the soil slope for

various confining pressures σ3. To estimate the saturated modulus of elasticity for each

soil layer of the slope, the confining pressure σ3 was determined as the overburden

pressure at the center of the investigated soil layer but not less than 9 kPa. Equation (3.1)

was then solved to calculate the unsaturated modulus of elasticity Eunsat associated with

the changes in matric suction. Once the matric suction changes and the associated

modulus of elasticity are estimated over time, the vertical soil movement can be predicted

at any time.

5.6.4 Prediction of vertical soil movement over time

To evaluate the vertical soil movement with time, the soil profile within the active zone

(i.e., 2.5 m) was divided into several layers with a 0.2 m thickness. The total vertical soil

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 169

movement at any point in the soil profile was calculated from equation (4.7). Figure

(5.33) shows the estimated vertical soil movements at different depths at the mid-slope

section compared with the field measurements.

Fig.5.33. Comparison of the estimated vertical soil movements with the field measurements at the mid-slope

Figure (5.33) shows, prior to the first artificial rainfall, there was no volume change taken

place in the slope since the initial matric suction was relatively high. After the

commencement of the first rainfall event, the high initial matric suction values

progressively reduced and the predicted soil heaves increased. After the first rainfall

stopped, the predicted soil heave terminated and a significant amount of shrinkage was

observed. Then, the predicted soil movements remained almost constant over the

remainder of the simulation period till they started increasing again as a response to the

second rainfall event.

The patterns of the predicted and the measured vertical soil movements in response to the

rainfall events were quite different. The average value of the percentage difference

between the predicted and the measured soil movements was 28%. It can be seen that,

after the commencement of the first rainfall event, the predicted heaves had a relatively

fast response to the rainfall infiltration compared to the measured values. This may be

CHAPTER 5 170

attributed to the difficulty to simulate the performance of the soil in the field considering

the high intensity of the initial soil cracks. In other words, the existence of cracks changes

the soil properties which, in turn, influence the soil-water interaction over time. However,

the results of modeling simulation were mainly based on uniform soil properties. In

addition, the field observations indicated that the soil cracks were large at the upper part

of the slope and decreased towards the lower part of the slope; nevertheless, the soil

layers and their properties were assumed in the simulation to be uniform over the entire

slope. Figure (5.33) also shows that the predicted soil heave terminated after the first

rainfall event and a significant amount of shrinkage appeared. However, the

measurements show soil heaves at a low rate over the period of the slope monitoring.

This can also be attributed to the uncertainty of the soil properties as the expansive soil

has intensive cracks in the field, and the difficulty to model its nature (soil cracks) and

response to the environmental changes.

In spite of the complexities associated with modeling the response of the natural

expansive soil deposit with cracks to the rainfall infiltration, the prediction of soil

movements improved when only the soil heave was considered (i.e., the soil

shrinkage was omitted) in the modeling (see Figure 5.34). Figure (5.34) shows that the

heave patterns from the predictions agreed with the field measurements. The average

percentage difference between the predicted and the measured soil movements after the

first rainfall event decreased to 21%. The results demonstrated that the MEBM could

be used with a reasonable degree of confidence for estimating the soil heave with

respect to time.

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 171

Fig.5.34. Comparison of the estimated soil heaves using the MEBM with the field measurements at the mid-slope

5.7 Case Study E: a Field Site in Arlington, Texas, USA (Briaud et al.

2003)

Case Study E is a site in Arlington, Texas, USA, selected for the validation of the MEBM

for predicting the field measurements of the movement of four full-scale spread footings

over a period of 2 years (1 August, 1999 to 31 October, 2001). Factors that influence the

volume change behavior of expansive soils, such as daily weather condition and

vegetation, were considered for this field construction site. The site was originally

monitored by Briaud et al. (2003) to investigate the damage caused by expansive soils in

both concrete and asphalt pavements, resulting in substantial discomfort, safety hazard,

and vehicle damage. The proposed MEBM is assessed more closely by comparing its

estimations of the soil movement at the considered site with the long term field

observations over 2 years. In addition, to provide more evidence to use the MEBM in

engineering practices, the results of the MEBM are compared with the estimated values

obtained from Briaud et al. (2003) and Zhang (2004) prediction methods presented in

Chapter Two.

CHAPTER 5 172

5.7.1 Site description

Figure (5.35) shows the soil stratigraphy, and the basic soil properties for the site. The

predominant soil type at the site is classified according to the Unified Soil Classification

System as borderline between CL and CH (Briaud et al. 2003). The stratigraphy of the

site consists of 0-1.8 m of dark gray silty clay and 1.8-4.0 m of brown silty clay. The soil

surface is covered by the Johnsongrass, which is warm season, perennial grass. The grass

root zone is suggested to be 0.22 m depth. The water level in the 7m deep standpipes

varies between 4 m and 4.8 m below the ground surface over 2 years.

Fig. 5.35. Soil stratigraphy and basic soil properties for the site of Case Study E (modified after Briaud et al. 2003)

The SWCC was measured by Briaud et al. (2003) using the pressure plate test and the salt

concentration test. Sigmaplot was used to obtain the regression curve of the all tests

results. Figure (5.36) shows the SWCCs and the mathematical expressions of the best

fitted SWCCs for the soil layers in the site. Permeability functions of soils under

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 173

unsaturated conditions are also required for the analysis. The soil permeability functions

were estimated by VADOSE/W using van Genucthen (1980) equation (see Figure 5.37).

(a) (b)

Fig. 5.36. Soil-water characteristic curves: (a) for dark gray silty clay, (b) for brown silty clay (modified after Briaud et al. 2003)

Fig. 5.37. Permeability functions estimated by VADOSE/W for the soil layers of Case Study E Figure (5.38) shows the daily temperature, relative humidity, wind speed, rainfall, and net

radiation over the period of study from 1 August, 1999 to 31 October, 2001. The daily

weather data was gathered from a weather station at the Arlington Municipal Airport

(Briaud et al. 2003, Zhang 2004).

CHAPTER 5 174

Fig. 5.38. Daily climate data over two years for the Arlington site of Case Study E (modified after Zhang 2004)

Four 10 m × 10 m areas were outlined at the site, and denoted as RF1, RF2, W1, and W2

as shown in Figure (5.35). Areas W1 and W2 were injected with 13.6 m3 of water/day for

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 175

3 days (6–8 July, 1999) at a depth of 3 m (Briaud et al. 2003). A 2 m square footing was

constructed at the center of each of the 10 m × 10 m areas between 15 and 30 July, 1999.

The vertical movements at the corners of each footing were recorded every month

starting from 11 August, 1999 until 1 November, 2001. The average values of the

measured movements for the four footings are shown in Figure (5.39).

Fig. 5.39. Measured movements of the footings at the Arlington site over two years (modified after Briaud et al. 2003)

5.7.2 Simulation of matric suction changes over time

The soil domain used for the simulation of matric suction changes over time using

VADOSE/W is 10 m × 10 m × 4 m ( length × width × height ) (Figure 5.40). A 10 m

length is considered to be far away so that there is no influence of the footing (Zhang

2004). The fully coupled transient analysis was conducted on the soil profile shown in

Figure (5.40) to estimate the matric suction changes over a period of 2 years.

VADOSE/W requires three categories of input parameters, namely: soil properties, initial

conditions, and boundary conditions. The soil properties included the SWCCs (Figure

5.36) and permeability functions (Figure 5.37). The variation of initial matric suction

versus depth shown in Figure (5.40) was provided by Zhang (2004) to represent the

initial condition of the simulation. The site soils were modeled using the simplified

CHAPTER 5 176

isothermal model since the fluctuation of the daily atmosphere temperature was

insignificant. Thus, the soil temperature, the thermal conductivity, and the volumetric

heat capacity were assumed to be constant and equal to 10 oC, 400 kJ/days/m/°C, and

1875 kJ/m³/°C, respectively. For nodes at the bottom boundary of the model domain, the

soil matric suction through the simulation was assumed to be constant and equal to -10

kPa (-1 m) considering the ground water level is about 4 m deep, and the temperature was

set up to be 10 oC (see Figure 5.40).

Fig. 5.40. The soil domain for Case Study E along with the initial and the boundary conditions used for the simulation of matric suction changes over time (modified after Zhang 2004)

The two year climate data (1 August, 1999 to 31 October, 2001) (Figure 5.38) was

applied on the surface layer around the footing as a climate boundary condition. A “no

flow” natural boundary condition was applied by default on the footing to represent the

footing as an impervious layer. At the ground surface outside from the footing, the soil is

covered by Johnsongrass (the most widely distributed naturalized warm-season, perennial

grass in North America); therefore, the boundary conditions are controlled by the

vegetation evepotranspiration. To mimic the in-situ condition, the vegetation was

considered to be an excellent grass with triangular distribution roots of 0.22 m deep.

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 177

Figure (5.41) shows the leaf area index function (LAI) of the excellent vegetation with a

maximum LAI value of 3 given in VADOSE/W (GeoSlope 2007), which was used for

this simulation. The plant moisture limiting point and the wilting point were assumed to

be 6000 kPa and 32000 kPa, respectively, as given by Zhang (2004).

Fig. 5.41. Typical leaf area index function for excellent grass coverage (modified after GeoSlope 2007)

To check the input data used in the simulation, the initial water content profile obtained

from VADOSE/W was compared with the initial water content presented in Zhang

(2004) (Figure 5.42). The good comparison between the initial values of water content

validates the simulation based on the used soil properties and boundary conditions. The

model was also validated against the field measurements. Figure (5.43) shows a close

agreement between the variations of the average values of the predicted and the measured

water contents for the four footings over the 3 m depth with respect to time (the average

percentage difference was 11%). This demonstrates that the soil model can be

successfully used for predicting the variation of matric suction over time, which is the

key information required for predicting the soil movement. Figure (5.44) shows the

matric suction profiles at the corner of the modeled footing predicted over the two-year

period. The matric suction profiles show significant variations near the ground surface

that decrease down to a depth where the variation becomes small.

CHAPTER 5 178

Fig. 5.42. Comparison between the initial water content profiles obtained from VADOSE/W and Zhang (2004)

Fig. 5.43. The variations of the average values of the predicted and the measured water contents for the four footings over the 3 m depth with respect to time

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 179

Fig. 5.44. Matric suction profiles at the corner of the footing simulated using VADOSE/W

5.7.3 Estimation of soil modulus of elasticity associated with matric suction

To estimate the modulus of elasticity associated with the changes in matric suction using

the VO model (Equation 3.1), µ is assumed to be 0.4 as suggested by Zhang (2004), β is

assumed to be 2 which is the recommended value for expansive soils, and α is assumed

to be 1/10 in order to provide a reasonable comparison between the predicted and

measured soil movements, and Esat is calculated based on the analysis of the oedometer

tests results given by Zhang (2004). Figure (5.45) shows the oedometer test results along

with their best-fitted curves (Equation 5.5) for the investigated soils, i.e., the dark gray

silty clay and the brown silty clay. Table (5.8) shows the fitting parameters of the void

ratio constitutive relationships shown in Figure (5.45). Once the constitutive

relationships of void ratio for the site soils are determined, the saturated modulus of

elasticity Esat can be calculated using equation (5.2). Figure (5.46) presents the variation

of the saturated modulus of elasticity with depth for the investigated soils at the Arlington

site (Case Study E).

CHAPTER 5 180

Table 5.8. Fitting parameters of the relationship of void ratio versus the mean stress

for the investigated soils at the Arlington site (Zhang 2004)

Soil material Fitting parameters a1 b1 x1 y1

Dark gray silty clay 0.49095 -0.42106 2.86157 0.19561 Brown silty clay 0.65549 -0.67522 3.48993 0.18089

Fig. 5.45. Oedometer test results and their fitting curves for the specimens of dark gray silty clay and brown silty clay at the Arlington site (modified after Zhang 2004)

Fig. 5.46. Variation of the saturated modulus of elasticity with depth for the investigated soils at the Arlington site

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 181

As a result, the unsaturated modulus of elasticity Eunsat associated with the matric

suction changes can be obtained from equation (3.1) in terms of the matric suction

changes (Figure 5.44), the SWCCs (Figure 5.36), and the saturated modulus of elasticity

values (Figure 5.46). The matric suction changes and the corresponding estimated

unsaturated modulus of elasticity are then used in conjunction with the volume change

constitutive relationship of soil structure to estimate the vertical soil movement over time.

5.7.4 Prediction of vertical soil movement over time

The 4 m depth of soil profile consisted of the two soils (i.e., Black gray silty clay and

brown silty clay) (Figure 5.40) was subdivided into ten layers. The thickness of each

layer was chosen such that the border between the two soils was located at a layer

boundary. The top two layers were suggested to have a thickness of 0.22 m (equivalent to

the thickness of the root depth) and 0.38 m, respectively, the bottom layer was assumed

to be with 0.6 m thickness, and the thickness of the rest layers were assumed to be 0.4 m.

The amount of soil movement for each soil layer associated with the change in matric

suction for each day was estimated using equation (4.6). The vertical soil movement of

each layer at a given time was calculated as a cumulated value of the soil movements for

all days prior to that given time. The total vertical soil movement at the ground surface

was obtained from the summation of the soil movements of all layers (Equation 4.7).

Figure (5.47) shows the predicted soil movements at the corner of the modeled footing

and the field measurements of the soil movement for the four footings. Compared with

the field measurements, the predicted soil movements reasonably matched the measured

movements in both tendency and magnitude over the first year. However, during the

second year, the predicted and the measured movements did not lead as good a match as

the values during the first year. It can be seen that the relatively steady evenly distributed

rainfall during the first year caused very small in situ movements, whereas the worst

drought in the history of the investigated area occurred during the second year caused

much large soil shrinkage in the field. The findings show that the proposed MEBM has

some limitations regarding its use to model the shrinkage behavior of expansive soils.

This can be attributed to the complexities associated with the mechanism of soil

CHAPTER 5 182

shrinkage, and the difficulty to quantify the influence of shrinkage cracks on the soil

movement.

Fig. 5.47. Comparison between the predicted soil movements at the corner of the modeled footing using the MEBM and the field measurements of the soil movement for the four footings

Review of Figure (5.47) shows that the predicted soil movements, on one hand, and the

measured soil movement at the footing RF1 (which doesn’t experience a large amount of

shrinkage), on the other hand, leads to a good comparison over the two years. The

maximum values of soil heave from both the prediction and the measurement agree well

and are approximately equal to 32 mm. The results demonstrate that the MEBM can be

used for estimating the soil heave over time with a reasonable degree of confidence.

Briaud et al. (2003) and Zhang (2004) used two different methods to predict the field

measurements of the footing movements over time at the Arlington site. The details of

both methods are presented in section 2.9.2. Figure (5.48) shows the comparison of the

predictions of soil movement using the MEBM, Briaud et al. (2003) method, and Zhang

(2004) method with the average field measurements of soil movement for each footing.

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 183

Fig. 5.48. Comparison of the soil movement predicted using different methods with the average field measurements of soil movement for the four footings

Figure (5.48) shows that, similar to the MEBM, Briaud et al. (2003) method reasonably

predicts the soil heave but not the soil shrinkage. Briaud et al. (2003) method is based on

the information of soil water content which is simpler and more reliable to measure in

comparison to the soil matric suction. However, it is uncoupled method where only the

influence of moisture variation on the soil volume change is considered. In addition,

Briaud et al. (2003) method requires a shrink test which is difficult to perform when the

soil is highly fractured. Another drawback is that any theoretical consideration must

make use of the SWCC to transform the equations from being suction-based method to

being water content-based method (Briaud et al. 2003).

Review of Figure (5.48) also shows that the predicted results of Zhang (2004) modeling

study don’t match the measurements of the soil movement for the four footings. The

modeled footing moved upward faster than the field measurements. This reflects that the

proposed approaches for modeling of grass root zone and for the construction of the

constitutive surfaces of the soil properties (void ratio, water content, degree of saturation,

permeability function) are not good enough for practical applications. For example, the

constitutive surfaces were constructed based on testing soils under conditions not

experienced in field, such as a shrinkage test at no normal stress, or consolidation test at

fully saturated conditions.

CHAPTER 5 184

5.8 Summary

The proposed Modulus of Elasticity Based Method (MEBM) is evaluated in this

chapter to estimate the vertical movement of natural expansive soils associated with

the changes in the environmental condition over time. The MEBM is based on the

theoretical concepts of unsaturated soils. It involves integrating the numerical modeling

results of the soil-atmospheric VADOSE/W program and the volume change constitutive

equation for unsaturated soils. The semi-empirical model proposed by Vanapalli and Oh

(2010) (VO model) was extended for unsaturated expansive soils and used as a tool to

estimate the variations of the modulus of elasticity with respect to matric suction. The

fitting parameters β = 2 and α = 0.05-0.15, as suggested in Chapter Three, were also used

in this chapter to provide reasonable estimations of the soil movement for the case studies

under consideration. The MEBM is a simple approach that requires only the matric

suction changes within the active zone depth and the associated modulus of elasticity to

predict the heave-shrinkage behavior of unsaturated expansive soils.

The performance of the MEBM was tested in five case studies originally investigated

in the literature by other researchers. The investigated case studies are representative

candidates of a variety of site conditions. Different scenarios with different initial and

boundary conditions were used to simulate each of the case studies. The results of the

MEBM reasonably agree with the published results (measurements/estimates) of the

case studies. The findings of this study demonstrate that the MEBM can be used with a

greater degree of confidence in engineering practice to predict the in situ expansive soil

movements with respect to time.

VALIDATION OF THE PROPOSED MODULUS OF ELASTICITY BASED METHOD 185

CHAPTER 6

ELASTICITY MODULI OF UNSATURATED EXPANSIVE

SOILS FROM DIMENSIONAL ANALYSIS

6.1 Introduction

The modulus of elasticity of unsaturated soils depends on numerous parameters such

as (i) the initial level of compaction (dry density, or void ratio), (ii) the initial state

hydration (water content, degree of saturation, or matric suction), and (iii) the

confinement (deviator stress, or lateral stress). The other factors that affect the

modulus of elasticity include variables such as boundary conditions, Poisson’s ratio,

specimen dimensions, soil structure (the size of soil particles), stress path, and stress

history, to list a few. The influence of all these parameters should be considered for a

reliable estimation of the soil modulus of elasticity. However, accounting the

influence of all these parameters requires extensive experimental programs and multi-

variable regression analyses. Such an approach is cumbersome to implement in the

conventional geotechnical engineering practice. Due to this reason, the modulus of

elasticity of unsaturated soils has been expressed in the literature as a function of only

one or two parameters. Different investigators such as Zhang et al. (2012) and Lu and

Kaya (2014) proposed a power function to quantitatively describe the relationship

between the soil modulus of elasticity and the soil water content. In Chapter Three,

the semi-empirical model proposed by Vanapalli and Oh (2010) (i.e., VO model) was

used for estimating the modulus of elasticity of unsaturated expansive soils as a

function of matric suction changes, neglecting the influence of mechanical stress

changes. Such an assumption is conservative and can be extended in practice for

pavements and lightly loaded residential structures, as per the earlier discussions

presented in Chapter Four. Some other investigators (e.g. Rahardjo et al. 2011) have

186

linked the modulus of elasticity to the change in both the net normal stress (i.e. the

mechanical stress) and the matric suction using multiple regression methods; this

approach is rigorous but it is time consuming from the view point of conducting

various experimental studies.

To alleviate some of the challenges associated with conducting cumbersome

experimental investigations to estimate the modulus of elasticity of unsaturated

expansive soils, dimensional analysis (hereafter referred to as DA) is used in this

chapter as a tool to propose an alternative approach. Some researchers (Butterfield

1999, Palmer 2008, Buzzi 2010, Buzzi et al. 2011) have used the DA in a few

geotechnical engineering applications; however, this approach is not widely used in

practice. One of the main advantages of the DA is that it allows intelligent

experiments; i.e., a reduction of the number of tests to be performed to characterize a

physical phenomenon taking account of the influence of all the parameters of an

activity or a phenomenon in the engineering applications. This is possible through the

use of dimensionless parameters, the number and form of which can be derived from

the Buckingham Pi theorem (Buckingham 1914).

For soils that are in a state of unsaturated condition, Matyas and Radhakrishna (1968),

Barden et al. (1969), Fredlund and Morgenstern (1977), Alonso et al. (1990), and

Gallipoli et al. (2003) proposed constitutive equations in terms of the soil void ratio

with respect to the changes associated with the net normal stress (i.e. the mechanical

stress) and the matric suction. Along similar lines, a dimensionless model extending

the DA is proposed in this chapter to estimate the modulus of elasticity for

unsaturated expansive soils, considering the influence of the state of hydration of soil

expressed in terms of the matric suction and the degree of saturation, the level of

compaction and the confinement described by the initial void ratio and the confining

stress, respectively. Experimental results of conventional and suction-controlled

triaxial tests for three expansive soils from Zao-Yang, Nanyang, and Guangxi in

China, published in Zhan (2003), Miao et al. (2002), and Miao et al. (2007),

respectively, that were used in Chapter Three for extending the VO model for

unsaturated expansive soils, are analyzed to form the dimensionless parameters ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 187

towards reliably estimating the soil modulus of elasticity. The validation of the

proposed dimensionless model is conducted by comparing its estimations of the

elasticity moduli with the experimental values of the elasticity moduli obtained from

the triaxial shear tests. The proposed dimensionless model is also verified using the

estimated values of the elasticity moduli from the semi-empirical model (i.e., VO

model) presented in Chapter Three.

In addition, the dimensionless model is also used as a tool in the modulus of elasticity

based method (MEBM) to estimate the modulus of elasticity of unsaturated expansive

soils for the most complicated case study (Case Study D) by Ng et al. (2003),

considering the influence of climatic conditions, soil properties, and soil cracks.

Comparisons are provided between the resulting soil movements and the field

measurements of this case study.

6.2 Dimensional Analysis Background

The dimensional analysis (DA) is a mathematical tool that shapes the general form of

relations that describe natural phenomena. The application of DA to any particular

physical phenomenon is based on the premise that the phenomenon can be described

by a list (V) of l variables (V1, V2,….., Vl), encompassing a total of m independent

primary dimensions (D) = (D1, D2, ….., Dm) (e.g. mass, length, time, temperature)

which is the minimum number of reference dimensions required to describe the

physical variables. The term ‘variables’ includes both the independent parameters of a

specific system (e.g. size, density, mass) and the dependent quantities such as

displacements, stresses, and bending moments (Butterfield 1999).

The objective of the DA is to minimize the dimension space in which the behavior of

specific system might be studied by combining assumed governing variables V into N

dimensionless parameters, N being less than V. In particular, Buckingham’s (1914)

theorem states that an initial equation involves l variables and m dimensions can

always be reduced to a dimensionless relationship involving only N dimensionless

parameters, where

CHAPTER 6 188

= −N l m (6.1)

The resulting N dimensionless parameters are conventionally labelled as

1 2( , ,........, )Nπ π π (i.e. Pi groups). As each π is dimensionless, the final function must

be dimensionless, and therefore dimensionally

0 0 01 2( , ,........, )Nf M L Tπ π π = (6.2)

The form of function f is not provided by the DA but it is usually approximated by an

empirical, dimensionless equation fitted to either model or prototype data. In addition, the

Buckingham theorem does not provide any specific guidance related to the choice of the

variables, which appear in each N parameter (i.e. Pi group) used for the reduction of the

problem. In order to enable systematic computation of dimensionless numbers, the

input and output variables of a concept are considered as performance variables. The

choice of repeating variables should be done within the concept’s internal variables

and according to the unique number of the system’s governing dimensions for best

results (Christophe et al. 2008).

6.3 Dimensional Analysis and Combination of Parameters

The dimensional analysis (DA) is used as a tool in this chapter to propose a

dimensionless model for estimating the modulus of elasticity of unsaturated expansive

soils. The application of DA to account for all of the factors influencing the value of the

modulus of elasticity of unsaturated expansive soils is challenging. This is due to

numerous properties and parameters that influence the modulus of elasticity of

unsaturated soils. In the present study, initial void ratio, matric suction, degree of

saturation, confining pressure, and deviator stress changes are assumed to be primary

factors. Experimental data resulting from the triaxial shear tests on unsaturated expansive

soils, performed under different confining stresses with varying matric suctions, are used

to apply the DA. Hence, the primary factors influencing the soil modulus of elasticity can

be expressed in terms of the initial height of soil specimen 0h , the change in specimen

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 189

height upon compression h∆ , the volume of voids voV , the volume of water wV , the

volume of solid particles sV , the matric suction ( )a wu u− , the confining stress 3σ , and

the change in deviator stress 1 3( )σ σ∆ − . The modulus of elasticity of unsaturated

expansive soils can be described using the list of these parameters as shown in equation

(6.3).

0 1 3 3( , , , , , ( ) , ( ) , ) 0vo s w a wf h h V V V u u σ σ σ∆ − ∆ − = (6.3)

Equation (6.3) involves 8 independent variables and 3 dimensions (mass, length, and

time). According to the Buckingham Pi theorem (Buckingham 1914), equation (6.3) can

be reduced to a simpler equation involving 5 dimensionless parameters (i.e., 8

independent variables – 3 dimensions) which depend on the combination of variables

present in equation (6.3). For a group of variables appearing in each dimensionless

parameter, the variables have to be combined in such a way that the powers of each of the

‘dimensions’ appearing in the group are separately zero. Several combinations are

possible to form dimensionless parameters. However, the correct sets of the

dimensionless parameters are needed to be selected and have to be verified through

experimental evidence. Equation (6.4) presents the used dimensionless parameters that

are chosen in this study based on obtaining most satisfactory agreement between the

experimental and the estimated values of the modulus of elasticity for the studied soils.

302

( )( ) ( )( , , , , ) 0unsat sat a a w a w

a a a

E E u u u u uf S eP P P

σ− − − −= (6.4)

where Eunsat and Esat are the elasticity moduli under unsaturated and saturated conditions,

respectively, S is the degree of saturation, 0e is the initial void ratio, and Pa is the

atmospheric pressure (i.e., 101.3 kPa) used for maintaining the parameters of the equation

(6.4) dimensionless. The DA is applied on experimental results of conventional and

controlled-suction triaxial tests available in the literature. The state of hydration of soil is

expressed in terms of the matric suction and the degree of saturation, the level of

compaction is described by the initial void ratio, and the confinement is described by the

confining stress. For simplicity and to significantly reduce the number of tests required,

CHAPTER 6 190

the four dimensionless parameters that are related to the degree of saturation, matric

suction, initial void ratio, and net confining stress for each of the tests are incorporated

into one unique dimensionless parameter X . This is more convenient as the equation to

estimate the modulus of elasticity of unsaturated soils can be reduced to a relationship

between only two entities. Using X allows accounting for the four influencing

parameters via only one dimensionless parameter, and equation (6.4) is modified to

( , ) 0unsat sat

a

E Ef XP−

= (6.5)

The parameter X can be defined in terms of several combinations of the dimensionless

parameters by a calibration procedure. The calibration in this study is achieved using a

program code for the suggested formula of X that incorporates the dimensionless

parameters with several exponents. The program code changes the exponents for the

formula incrementally and calculates the soil modulus of elasticity. The calibrated

formula of X is determined on the basis of the best agreement between the experimental

and the estimated values of the soil modulus of elasticity. Based on the results of triaxial

tests for the three unsaturated expansive soils from Zao-Yang, Nanyang, and Guangxi in

China, presented in Chapter Three, the following relationship is suggested to be the

calibrated formula of X .

0.1 0.732

0

( ) ( ) ( )1( )[( ) ( )]a a w a w

a a

u u u u uX Se P P

σ − − −= + (6.6)

The relation of the exponents of equation (6.6) with other factors influencing the soil

modulus of elasticity under unsaturated condition is not clear. The exponents of the

equation could depend on soil structure, mechanical history, and other soil properties. No

unique or well defined relationship could be derived despite attempts to correlate the

values of the exponents to the soil properties reported from the three discussed studies

(Zhan 2003, Miao et al. 2002, Miao et al. 2007).

According to the DA, equation (6.5) can be written in terms of the dimensionless

parameter X as

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 191

( )unsat sat

a

E E f XP−

= (6.7)

The only manner to assess the proposed dimensionless model (Equation 6.7) is to use

experimental data and plot the results in terms of X versus ( )unsat sat aE E P− . The results

of triaxial tests presented in Chapter Three for three different expansive soils are used for

examining the validity of equation (6.7). The values of elasticity moduli of soils in the

first entity of equation (6.7) are experimentally determined from the stress-strain curves

of triaxial tests during shearing of saturated/unsaturated compacted specimens under

different confining stresses and matric suctions. As previously discussed in Chapter

Three, the experimental values of modulus of elasticity are determined as the reciprocal

of intercept of the straight lines resulting from plotting the stress-strain relationships on

the transformed axes ε and 1 3/ ( )ε σ σ− (Figure 3.4). If a good correlation with a reasonable

coefficient of determination R2 could be found between the two entities X and

( )unsat sat aE E P− , equation (6.7) would be used to back-calculate the soil modulus of

elasticity at any unsaturated condition.

The above analysis has been extended, and comparisons are provided between the values

of the modulus of elasticity derived from the triaxial tests results and the proposed

dimensionless model (Equation 6.7) to check its capability for estimating the modulus of

elasticity of expansive soils.

6.4 Triaxial Tests Results Used in the Dimensional Analysis

The three data sets from Zhan (2003), Miao et al. (2002), and Miao et al. (2007) for

compacted expansive soils from Zao-Yang, Nanyang, and Guangxi in China,

respectively, that were analyzed in Chapter Three, are also used in the dimensional

analysis. The key soil properties and the results of conventional and suction-controlled

triaxial tests on saturated and unsaturated compacted specimens of these three soils under

different confining stresses and matric suctions are presented in Chapter Three. The stress

versus strain relationships of the triaxial tests were analyzed and plotted on the

CHAPTER 6 192

transformed axes ε and 1 3/ ( )ε σ σ− . The straight line equation (3.6) was used to fit the

data. The experimental values of soil modulus of elasticity were determined as the

reciprocal of the intercepts of the resulting straight lines. The experimental values of

elasticity moduli for Zao-Yang, Nanyang, and Guangxi expansive soils under saturated

and unsaturated conditions are summarized in Table (3.2), (3.3), and (3.4), respectively.

6.5 The Application of Dimensional Analysis for Estimating the Soil

Modulus of Elasticity

The first step to apply the DA approach for estimating the modulus of elasticity of

unsaturated expansive soils involves calculating the dimensionless parameter X using

equation (6.6). The data required to calculate X are available for the three soils (i.e.,

Zao-Yang, Nanyang, and Guangxi expansive soils), which include the initial void ratio

0e , the net confining stress 3( )auσ − , the matric suction ( )a wu u− , and the degree of

saturation S . The second step involves plotting X versus ( )unsat sat aE E P− obtained from

the results of triaxial tests. A good correlation between the two entities X and

( )unsat sat aE E P− validates the proposed DA approach for estimating the modulus of

elasticity of unsaturated expansive soils. By using the regression analysis, the best fit

equation can be found for the relationship between X and ( )unsat sat aE E P− (Equation

6.7) (i.e., the dimensionless model).

Seven controlled-suction triaxial shear tests conducted by Zhan (2003) on unsaturated

compacted specimens of Zao-Yang soils, the results of which are available in Zhan

(2003), were used in the DA. First, the dimensionless parameter X was calculated using

the tests data shown in Table (6.1). Then, plotting the parameter X versus

( )unsat sat aE E P− for unsaturated specimens tested under each net confining stress led to

satisfactory correlations; the coefficients of determination R2 = 0.98 and 0.97 were

obtained for the soil specimens tested under 3( )auσ − = 50 kPa and 200 kPa, respectively

(Figure 6.1). In the present analysis, hyperbolic models were chosen to fit the data;

however, other trends could be used if they lead to a higher coefficient of determination.

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 193

The results suggest that the proposed DA can be successfully applied on the triaxial tests

results of Zao-Yang soils.

Table 6.1. Experimental data and the corresponding dimensionless parameter X for the

compacted specimens of Zao-Yang soils under unsaturated conditions (data from Zhan

(2003))

(σ3 – ua) (kPa)

(ua – uw) (kPa)

e0 S X

50 25 0.779 0.783 0.63 50 50 0.764 0.745 1.04 50 100 0.746 0.701 1.73 50 200 0.730 0.685 2.87 200 25 0.698 0.866 1.15 200 100 0.676 0.769 3.12 200 200 0.699 0.707 4.90

Fig. 6.1. The relationship of the dimensionless parameter X versus ( )unsat sat aE E P− for compacted specimens of Zao-Yang soils tested under different confining stresses

Equations (6.8) and (6.9) represent the hyperbolic models that have been used to fit the

relationships of X versus ( )unsat sat aE E P− for the specimens tested under confining

stress 3( )auσ − of 50 and 200 kPa, respectively.

CHAPTER 6 194

0.65267.43( )unsat sat

a

E E XP−

= (6.8)

1.6120.15( )unsat sat

a

E E XP−

= (6.9)

Equations (6.8) and (6.9) have been used to back-calculate the modulus of elasticity of

Zao-Yang soil under unsaturated condition Eunsat for 3( )− auσ = 50 kPa and 200 kPa,

respectively. Figure (6.2) shows the comparisons between the experimental and the

estimated values of the unsaturated modulus of elasticity Eunsat. A good agreement was

observed between the elasticity moduli obtained from the experimental results and the

proposed dimensionless model (R2 = 0.98 for 3( )− auσ = 50 kPa, and R2 = 0.92 for

3( )− auσ = 200 kPa).

Fig. 6.2. Comparison between the experimental and the estimated values of the unsaturated modulus of elasticity for Zao-Yang soil

The proposed DA approach was also extended for the triaxial tests carried out by Miao et

al. (2002) on the compacted specimens of Nanyang soil. The initial void ratio, the degree

of saturation, the matric suction, and the net confining stress for each test are provided in

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 195

Table (6.2). The dimensionless parameter X was calculated for each test using equation

(6.6). The experimental values of elasticity moduli of Nanyang soil summarized in Table

(3.3) were used to calculate ( )unsat sat aE E P− . The relationship of X versus

( )unsat sat aE E P− was plotted for each confining stress (i.e., each group) as shown in

Figure (6.3). Good correlations were found between X and ( )unsat sat aE E P− for the

series of the tests with the exception of the tests group of 112.5 kPa net confining stress.

This set of data appears to be inconsistent with the remainder of the results (the tests

groups of 25 kPa and 62.5 kPa net confining stress). This may be attributed to

considering the average value of the net confining stresses to represent a group of tests, or

to a measuring error during some tests which leads to no change in the modulus of

elasticity as a response of a change in matric suction (see Table (3.3)).

Table 6.2. Experimental data and the corresponding dimensionless parameter X for the

compacted specimens of Nanyang soil under unsaturated conditions (data from Miao et

al. (2002))

(σ3 – ua) (kPa)

Average (σ3 – ua) (kPa)

(ua – uw) (kPa)

e0 S X

30 25 50 0.8 0.71 0.87 20 80 0.8 0.68 1.21 30 120 0.8 0.65 1.60 20 200 0.8 0.61 2.29 50 62.5 50 0.8 0.71 1.05 70 80 0.8 0.68 1.46 80 120 0.8 0.65 1.93 50 200 0.8 0.61 2.76 100 112.5 50 0.8 0.71 1.27 120 80 0.8 0.68 1.77 130 120 0.8 0.65 2.34 100 200 0.8 0.61 3.34

The relationships between X and ( )unsat sat aE E P− for the specimens of Nanyang soil

tested under average values of 25 kPa and 62.5 kPa confining stress (Figure 6.3) can be

expressed using the best fitting equation (6.10) and equation (6.11), respectively.

CHAPTER 6 196

1.53114.09( )unsat sat

a

E E XP−

= (6.10)

1.0096.08( )unsat sat

a

E E XP−

= (6.11)

Fig. 6.3. The relationship of X versus ( )unsat sat aE E P− for compacted specimens of Nanyang soil tested under different confining stresses

Equations (6.10) and (6.11) were used to back-calculate the elasticity moduli of Nanyang

soil under unsaturated conditions Eunsat. Figure (6.4) presents the comparison between the

experimental and estimated values of unsaturated modulus of elasticity Eunsat. The

coefficients of determination were reasonable (R2 = 0.92 for 3( )auσ − = 25, and R2 = 0.8

for 3( )auσ − = 62.5 kPa). It is concluded that the unsaturated modulus of elasticity at a

given confining stress for the compacted specimens of Nanyang soil can be reasonably

estimated using the DA. However, the applicability of the DA to the tests group of 112.5

kPa net confining stress requires further investigation.

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 197

Fig. 6.4. Comparison between the experimental and the estimated values of the unsaturated modulus of elasticity for Nanyang soil

Other data sets used to validate the proposed dimensionless model for reproducing Eunsat

were given by Miao et al. (2007) (i.e., Guangxi soil). The data shown in Table (6.3) were

used to calculate the dimensionless parameter X (Equation 6.6). Figure (6.5) shows

( )unsat sat aE E P− , calculated in terms of the experimental values of saturated/unsaturated

modulus of elasticity of Guangxi soil, versus X for each set of triaxial tests conducted

under a certain net confining stress. Equations (6.12–6.14) represent the relationships

between X and ( )unsat sat aE E P− for the data sets under 50, 100, and 200 kPa of net

confining stress, respectively.

2.6210.74( )unsat sat

a

E E XP−

= (6.12)

1.6719.78( )unsat sat

a

E E XP−

= (6.13)

0.6654.11( )unsat sat

a

E E XP−

= (6.14)

CHAPTER 6 198

Table 6.3. Experimental data and the corresponding dimensionless parameter X for the

compacted specimens of Guangxi soil under unsaturated conditions (data from Miao et

al., 2007)

Fig. 6.5. The relationship of X versus ( )unsat sat aE E P− for compacted specimens of Guangxi soil tested under different confining stresses

In Figure (6.5), the coefficients of determination of 0.97, 0.98, and 0.99 were obtained for

specimens tested under the net confining stress of 50, 100, and 200 kPa, respectively. The

excellent correlation provides a greater degree of confidence for the applicability of the

DA approach to the experimental data of Guangxi soil.

(σ3 – ua) (kPa)

(ua – uw) (kPa)

e0 S X

50 179.37 0.824 0.76 2.37 50 121.81 0.824 0.85 1.81 50 57.37 0.824 0.92 1.08 100 179.37 0.824 0.76 2.90 100 121.81 0.824 0.85 2.22 100 57.37 0.824 0.92 1.31 200 179.37 0.824 0.76 3.86 200 121.81 0.824 0.84 2.95 200 57.37 0.824 0.92 1.74

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 199

Figure (6.6) presents the comparison between the experimental values of Eunsat derived

from the triaxial tests and those back-calculated using equations (6.12–6.14) for 3( )auσ −

= 50, 100, and 200 kPa, respectively. A close agreement was obtained between the values

of the elasticity moduli. The correlation was relatively high (R2 = 0.92, 0.97, and 0.99).

Consequently, the Eunsat of the Guangxi soil compacted at a given net confining stress can

be reliably estimated using the proposed dimensionless model.

Fig. 6.6. Comparison between the experimental and estimated values of the unsaturated modulus of elasticity for Guangxi soil

6.6 Verification of the Proposed Dimensionless Model

The verification of the proposed dimensionless model was assessed by comparing its

estimations of the soil modulus of elasticity with those obtained from the VO model

presented in Chapter Three. The VO model (Equation 3.1) was proposed by Vanapalli

and Oh (2010) for the estimation of the unsaturated modulus of elasticity of both coarse

and fine-grained soils with plasticity index Ip < 16%. In Chapter Three, the VO model

was extended for unsaturated expansive soils (i.e., Ip > 16%).

( )1/101.3a w

unsat sata

u uE E S

Pβα

− = +

(3.1)

CHAPTER 6 200

where Eunsat and Esat is soil modulus of elasticity under unsaturated and saturated

condition, respectively, ( )a wu u− is matric suction, Pa is atmospheric pressure (Pa =

101.3 kPa), and S is degree of saturation.

The values of the unsaturated modulus of elasticity Eunsat for the three expansive soils

(Zao-Yang, Nanyang, and Guangxi expansive soils) were reasonably estimated using the

VO model (Equation 3.1) as presented in Chapter Three. The fitting parameters β = 2 and

α = 0.05−0.15 were found to provide the elasticity moduli for the three expansive soils

that reasonably agree with the values of the elasticity moduli obtained from the triaxial

shear tests (R2 = 0.91) (see Figure 3.18).

Figure (6.7) presents the comparison of the values of Eunsat for the three expansive soils

obtained from the proposed dimensionless model and the VO model. It can be seen that

the dimensionless model provides a good agreement with the estimations of Eunsat from

the VO model. The coefficient of determination was relatively high (R2 = 0.9). As a

result, the proposed dimensionless model can be used for estimating the modulus of

elasticity of unsaturated expansive soils under any net confining stress (i.e. any loading

condition) with varying matric suctions.

Fig. 6.7. Comparison between the values of elasticity moduli estimated from the proposed dimensionless model and the VO model for the three investigated expansive soils

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 201

The VO model (Equation 3.1) neglects the influence of the mechanical stress on the

modulus of elasticity of unsaturated expansive soils. Such an assumption is conservative

and can be extended in practice for pavements and lightly loaded residential structures,

where the influence of net normal stress is insignificant and only matric suction changes

have a predominant influence on the soil volume change. However, compared with the

results of the VO model, the closer agreement between the Eunsat values from the

dimensionless model and the triaxial tests results with higher coefficient of determination

(R2 = 0.97) (Figure 6.8) suggests that the dimensionless model (Equation 6.7) can be used

with greater confidence than the VO model (Equation 3.1). In other words, the proposed

dimensionless model is more rigorous and reliable, and it can be used for all scenarios of

loading conditions (both lightly and heavily loaded structures) for the estimation of

modulus of elasticity for unsaturated expansive soils.

Fig. 6.8. Comparison between the values of elasticity moduli obtained from the dimensionless model and the triaxial tests for the three investigated expansive soils

6.7 Prediction of Vertical Movement of Unsaturated Expansive Soils

Based on the Dimensionless Model

The modulus of elasticity based method (MEBM) has been used in this research study for

predicting the long-term vertical movements of unsaturated expansive soils considering

CHAPTER 6 202

the environmental factors. The predictions can be made using only the data of initial

matric suction, and the SWCC and the saturated modulus of elasticity measured from

fairly routine geotechnical laboratory tests. In the MEBM, the VO model (Equation 3.1)

has been used to obtain the unsaturated modulus of elasticity as a function of matric

suction. The VO model can be reliably used in practice for pavements and lightly loaded

residential structures. However, to conduct a reliable estimation of the soil movements, it

is often desirable to effectively describe the soil modulus of elasticity as a function of all

its influencing parameters. In this chapter, a new dimensionless model has been

successfully proposed and used for estimating the modulus of elasticity of unsaturated

expansive soils, taking into account the effect of the matric suction, the net confining

stress, the initial void ratio, and the degree of saturation. The dimensionless model can be

used for lightly and heavily loaded structures with a greater degree of confidence.

Case Study D by Ng et al. (2003) previously simulated in Chapter Five is revisited in this

section to evaluate the MEBM approach extending the proposed dimensionless model for

estimating the modulus of elasticity. Case Study D is chosen here because its triaxial tests

results were analyzed dimensionally earlier in this chapter for estimating the unsaturated

modulus of elasticity. Equations (6.8) and (6.9) are the dimensionless models proposed to

estimate the unsaturated modulus of elasticity under confining stresses of 50 kPa and 200

kPa, respectively.

The exact value of the confining stress applied in the field is required to accurately

estimate the values of the soil modulus of elasticity at any depth. However, the triaxial

tests conducted by Ng et al. (2003) on compacted soil specimens for Case Study D were

limited to certain values of the confining stress, and do not represent the in-situ condition.

To evaluate the soil movements for the case study based on using the proposed

dimensionless model for estimating the soil modulus of elasticity, it has been assumed

that: i) the dimensionless model developed for specimens tested under the confining

stress of 50 kPa (Equation 6.8) can be used to estimate the modulus of elasticity at the

middle of the soil layers considered for Case Study D; ii) the confining stress at any depth

can be calculated as the overburden pressure at that depth.

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 203

The same soil profile of Case Study D with the 2.5 m active zone depth, simulated in

Chapter Five, are used here to predict the vertical soil movements over 30 days based on

using the proposed dimensionless model (Equation 6.8). The daily changes in matric

suction simulated by VADOSE/W and the corresponding unsaturated modulus of

elasticity estimated by the dimensionless model are substituted into the soil movement

constitutive relationship (Equation 4.7). Figure (6.9) shows the predicted and the

measured soil movements (heave/shrink) at the mid-slope section R2 at different

depths.

Fig. 6.9. Comparison of the predicted soil movements using the MEBM based on the dimensionless model with the field measurements at the mid-slope

The results show that the MEBM based on the proposed dimensionless model predicts

the patterns of the in situ soil movements. However, the values of the predicted

movements near the ground surface (0.1 m and 0.5 m) are very small compared with

the field measurements. This can be attributed to the assumptions suggested for

estimating the unsaturated modulus of elasticity as discussed before. In addition, the

elasticity moduli estimated by the dimensionless model, that is developed based on

the results of the triaxial tests for compacted specimens, are expected to have higher

values compared with the elasticity moduli obtained for undisturbed specimens of soil

that have cracks and fissures in the field.

CHAPTER 6 204

6.8 Summary

The dimensional analysis (DA) was successfully used as a tool to propose a

dimensionless model for estimating the modulus of elasticity of unsaturated expansive

soils. The proposed model takes into account the effect of matric suction and net

confining stress along with the initial void ratio and the degree of saturation towards

comprehensive characterization of the unsaturated modulus of elasticity. The validation

of the proposed dimensionless model was conducted using the experimental results of

conventional and suction-controlled triaxial tests for the three expansive soils, namely

Zao-Yang, Nanyang, and Guangxi expansive soils. A good correlation with a high

determination coefficient (R2 = 0.97) was obtained between the values of the unsaturated

modulus of elasticity obtained from the dimensionless model and the triaxial tests results.

However, more triaxial tests on expansive soils under different confining stresses and

matric suctions are required to calibrate the fitting parameters of the dimensionless

model.

The values of unsaturated modulus of elasticity Eunsat for the three investigated expansive

soils estimated by the VO model were used to verify the proposed dimensionless model.

The dimensionless model provided a greater degree of confidence for estimating the

modulus of elasticity of unsaturated expansive soils compared with the results of the VO

model. The VO model can be reliably used in practice for pavements and lightly loaded

residential structures, while the proposed dimensionless model can be used for all

scenarios of loading conditions. The proposed dimensionless model requires soil

properties that can be determined from a limited number of routine laboratory tests.

Case Study D investigated by Ng et al. (2003) was revisited for the evaluation of the

MEBM approach based on using the dimensionless model for estimating the modulus of

elasticity. It is expected that using this innovative dimensionless model in the MEBM

will provide more reliable predictions of the heave and shrink behavior of expansive soils

if a limited number of triaxial shear tests can be conducted on undisturbed specimens

collected from the active zone depth.

ELASTICITY MODULI OF UNSATURATED EXPANSIVE SOILS FROM DIMENSIONAL ANALYSIS 205

CHAPTER 7

CONCLUSIONS AND FUTURE RESEARCH SUGGESTIONS

7.1 Introduction

The overall objective of this thesis is to develop a general and simple method for

predicting the vertical movement of unsaturated expansive soils considering soil-

atmospheric interactions within the active zone. This method is referred to as the modulus

of elasticity based method (MEBM). The changes in matric suction and the associated

modulus of elasticity are the key parameters required in the MEBM for estimating the

vertical soil movements. The MEBM was validated for five different case studies from

three countries: Canada, China, and the United States. These case studies are referred as

Case Study A (Vu and Fredlund 2006), Case Study B (Yoshida et al. 1983, Vu and

Fredlund 2004), Case Study C (Ito and Hu 2011), Case Study D (Ng et al. 2003), and

Case Study E (Briaud et al. 2003). Several scenarios along with different boundary

conditions were used to simulate each of these case studies. The step-by-step procedure

of the MEBM includes: (i) simulation of the matric suction variations over time; (ii)

estimation of the corresponding modulus of elasticity of the matric suction value; (iii)

prediction of the vertical soil movements with respect to time.

The results of the research program are valuable to provide guidelines for rational design

of lightly loaded structures placed in/on expansive soils using the mechanics of

unsaturated soils. The MEBM approach is simple in comparison to other presently

available methods and can be used in conventional geotechnical engineering practice.

In the following sections, conclusions derived from this research study are summarized.

In addition, suggestions for future research with respect to the estimation of the vertical

206

movement related volume change along with other applied research for expansive soils

are highlighted.

7.2 Conclusions

7.2.1 Overall performance of the MEBM approach

• The MEBM is a straightforward and simple approach that has been

successfully used to simulate the heave and/or shrink related volume change

of expansive soils with respect to time within the active zone. The MEBM

has been accurately validated for different case studies published in the

literature:

- For Case Study A, a slab-on-ground placed on Regina expansive clay

subjected to a constant infiltration rate over 175 days, the 1-D heave of

the unsaturated expansive soil deposit during the infiltration were

successfully predicted at different depths using the MEBM. The

coefficient of determination between the results of the MEBM and the

numerical modeling results of Vu and Fredlund (2006) for Case Study

A was relatively high (R2 = 0.97).

- For Case Study B, a light industrial building constructed on Regina

expansive clay in Saskatchewan, Canada, the soil heaves due to a

water leak below the floor slab which were predicted using the MEBM

agreed well with the data (measurements/estimates) published by Vu

and Fredlund (2004) over 150 days (R2 > 0.94).

- For Case Study C, the other test site in Regina, Saskatchewan, Canada,

the factors influencing the soil movements such as soil cracks, cover

type (pavement/vegetation), lawn irrigation, climate conditions, and

vegetation were successfully considered over one year (1 May,

2009−30 April, 2010). The total soil movement (i.e., the difference

between the maximum values of soil heave and shrinkage) predicted

CONCLUSIONS AND FUTURE RESEARCH SUGGESTIONS 207

using the proposed MEBM was close to that predicted by Ito and

Hu (2011) with a 6% difference.

- For Case Study D, a cut-slope in an expansive soil in Zao-yang, Hubie,

China, the MEBM was validated against the field measurements of the

soil movements over one month (13 August−12 September, 2001),

considering the effect of climatic conditions, two artificial rainfall

events, and soil cracks. The average percentage difference between the

predicted and the measured soil movements was 28%. The prediction

of soil movements was improved when only the soil heave was

considered. The average percentage difference decreased to 21%, and

the heave patterns from the prediction were similar to the field

measurements.

- For Case Study E, a field site in Arlington, Texas, USA, the predicted

movements of the four full-scale spread footings in the site matched

the measured movements reasonably well in both tendency and

magnitude for the first year. However, the predicted and the measured

movements during the second year did not lead as good a match as the

values during the first year. This is due to the complexities associated

with the mechanism of soil shrinkage, and the difficulty to quantify the

influence of shrinkage cracks on the soil movement.

• The volume change constitutive equation of the MEBM approach was

developed based on the assumption that the mechanical stress remains

constant during the heave/shrinkage processes. The assumption is not strictly

valid as the soil density changes due to a couple of factors. However, for

lightly loaded structures, where the MEBM can be applied, the influence of

the mechanical stress is insignificant in several scenarios and can be

neglected. Such an assumption is also conservative and can be extended in

practice. Furthermore, the volume change of expansive soils associated with

CHAPTER 7 208

the variations of environmental conditions occurs near the ground surface and

decreases with depth.

• The matric suction variations and the corresponding values of the modulus of

elasticity along with the soil-water characteristic curve (SWCC) of soils are

the most important parameters that contribute to the swelling and shrinkage

behavior of expansive soils. The swelling capacity of soil is essentially

dependent on the elastic properties of the solid phase and caused by the

expansive soil’s affinity for water. The strength of the MEBM model lies in

its use of the soil properties that can be determined by using conventional

geotechnical testing methods.

7.2.2 Soil-atmospheric interaction

• Estimation of the soil matric suction as a function of time using the soil-

atmospheric interaction model (VADOSE/W) allows for the variations of soil

profile characteristics, the water infiltration/migration, and the variations of

climatic conditions to be taken into account.

• The results of the VADOSE/W analyses demonstrated that rigorous soil-

atmospheric interaction modeling can be performed to estimate the time

evolution of matric suction profile and the depth of the active zone. For

example, a reasonable agreement was observed between the matric suction

values obtained from VADOSE/W and those estimated by Vu and Fredlund

(2004). The coefficient of determination was relatively high (R2 > 0.89).

• The results of VADOSE/W simulations for the five case studies show that the

environmental conditions will primarily influence the surface layer of the soil

profile, which constitutes the active zone depth that is typically 2 m to 3.5 m.

The matric suction profiles were found to vary with depth and time, and

correlated well with the environmental conditions on the surface

boundary.

CONCLUSIONS AND FUTURE RESEARCH SUGGESTIONS 209

7.2.3 Unsaturated modulus of elasticity

• The VO model (i.e., Vanapalli and Oh (2010) model) with two fitting

parameters α and β was extended in this study for expansive soils to estimate

the variations of the modulus of elasticity with respect to matric suction. The

two fitting parameters β = 2 and α = (0.05−0.15) were recommended for

modeling the volume change behaviour of expansive soils. The fitting

parameter β = 2 was found to be suitable for the three investigated expansive

soils (i.e., Zao-Yang, Nanyang, and Guangxi expansive soils). The value of α

was defined on the basis of the best agreement between the experimental and

the predicted values of the modulus of elasticity with respect to matric

suction (R2 = 0.77−0.97).

• In spite of the susceptibility of the measured modulus of elasticity to

measurement errors or problems associated with the experimental techniques

and the difficulties of ensuring that the stress path was entirely elastic, the

adopted VO model reasonably predicted the modulus of elasticity obtained

from the experimental data of triaxial tests for the three unsaturated

expansive soils studied (R2 = 0.91).

• Reasonable predictions of the vertical soil movements over time for five case

studies (i.e., Case Studies A, B, C, D, and E) were achieved by using the

proposed MEBM extending the VO model for estimating the unsaturated

modulus of elasticity. For all the case studies β was equal to 2 and the α

value was between 0.05−0.15. The VO model can be reliably used in practice

for pavements and lightly loaded residential structures.

• A dimensionless model for estimating the modulus of elasticity of unsaturated

expansive soils was successfully developed using the dimensional analysis

(DA). This dimensionless model accounts for the effect of the matric suction

and the net confining stress, along with the initial void ratio and the degree of

saturation. A good correlation with a high determination coefficient (R2 =

CHAPTER 7 210

0.97) was obtained between the values of the unsaturated modulus of elasticity

obtained from the proposed dimensionless model and from the results of

triaxial shear tests for the three expansive soils studied. The dimensionless

model can be used in practice for all scenarios of loading conditions (both

lightly and heavily loaded structures).

7.3 Recommendations and Suggestions for Future Research Studies

• The MEBM proposed in this thesis research is validated and tested in five case

studies. More comprehensive field studies from different regions of the world are

necessary in order to provide more evidence for the use of the MEBM in

engineering practices.

• In addition to matric suction changes, it is also important to consider the influence

of other parameters, such as overburden pressure, soil cracks, and soil

temperature, in the numerical modeling solutions of the soil-atmospheric

interactions which are subsequently used for predicting the soil movement over

time.

• Changes in the soil volume as soil suction is increased/decreased can significantly

affect the interpretation of soil-water characteristic curve information and result in

erroneous calculations of the unsaturated soil property functions (e.g., soil

permeability function and degree of saturation) (Fredlund and Houston 2013).

However, the SWCC has been treated in this study as a single approximate

relationship between the amount of water in a soil and soil suction. It will be very

interesting to properly account for the effects of volume change on the SWCC of

unsaturated expansive soils.

• Conducting a sensitivity analysis or a parametric study of the parameters that

affect the volume change behavior of expansive soils would be valuable. Some of

the key parameters include the saturated modulus of elasticity, the saturated

CONCLUSIONS AND FUTURE RESEARCH SUGGESTIONS 211

coefficient of permeability, Poisson’s ratio, the swelling index, and the

number/thickness of soil layers.

• The fitting parameters of the VO model used in this research study are found to

provide reasonable values for the modulus of elasticity which are successfully

used to reproduce the in situ soil movements. However, more tests for expansive

soils of various plasticity index values are needed to provide a generalized

relationship of the fitting parameters in terms of the conventional soil properties

such as the plasticity index, Ip.

• The dimensionless model is developed based on a limited number of the

conventional and suction-controlled triaxial tests for three different expansive

soils. However, more triaxial test results for the investigated soils under the same

values of the confining stresses used in this study but with different matric

suctions are necessary to calibrate the fitting parameters of the proposed

dimensionless model for each soil.

• Case Study D has been revisited for the evaluation of the MEBM approach based

on using the innovative dimensionless model as a tool for estimating the soil

modulus of elasticity. It is expected that using the dimensionless model in the

MEBM would provide more reliable predictions of heave and shrinkage behavior

of expansive soils if the triaxial shear tests were conducted on undisturbed

specimens collected from the field.

• Using an easy and simple way to obtain high quality data for the estimation of the

modulus of elasticity of unsaturated expansive soils can further improve the

prediction results of the MEBM.

CHAPTER 7 212

REFERENCES

Abbaszadeh, M. 2011. The effect of cracks on unsaturated flow and volume change

properties of expansive clays and impacts on foundation performance. Ph.D.

Thesis. Arizona State University, Tempe, Arizona.

Abdullah, W.S. 2002. Bidemsional swell effect on accuracy of footing heave prediction.

Geotechnical Testing Journal, 25(2): 177–186.

Aitchison, G.D. 1960. Relationships of moisture stress and effective stress functions in

unsaturated Soils. In Proceedings of the Conference on Pore Pressure and Suction

in Soils, Butterworths, London, UK, pp. 47–52.

Aitchison, G.D. 1965. Discussion in Proceedings of the Sixth International Conference

on Soil Mechanics and Foundation Engineering, Montreal, QC, pp. 318–321.

Aitchison, G.D. 1973. The quantitative description of the stress-deformation behavior of

expansive soils – preface to set of Papers. In Proceedings of the Third International

Conference on Expansive Soils, Haifa, pp. 79–82.

Aitchison, G.D. and Martin, R. 1973. A membrane oedometer for complex stress-path

studies in expansive clays. In Proceedings of the Third International Conference on

Expansive Soils, Haifa, pp. 83–88.

Aitchison, G.D. and Woodburn, J.A. 1969. Soil suction in foundation design. In

Proceedings of the Seventh International Conference on Soil Mechanics and

Foundation Engineering, Mexico City.

Aitchison, G.D., Peter, P. and Martin, R. 1973. The quantitative description of the stress

deformation behaviour of expansive soils. CSIRO Division of Applied

Geomechanincs, Research Paper.

213

Akbas, S.O. and Kulhawy, F.H. 2009. Axial compression of footings in cohesionless

soils. I: load settlement behavior. Journal of Geotechnical and Geoenvironmental

Engineering, ASCE, 123(11): 1562–1574.

Allam, M.M. and Sridharan, A. 1987. Effect of repeated wetting and drying on shear

strength. Journal of the Geotechnical Engineering, ASCE, 107: 121–438.

Allman, M.A., Delaney, M.D. and Smith, D.W. 1998. A field study of seasonal ground

movements in expansive soils. In Proceedings of the Second International

Conference on Unsaturated Soils, International Academic Publishers, Beijing, pp.

309–314.

Alonso, E.E., Gens, A. and Hight, D.W. 1987. Special problem soils. In Proceedings of

the Ninth European Conference on Soil Mechanics and Foundation Engineering,

Dublin, pp. 1087–1146.

Alonso, E.E., Gens, A. and Josa, A. 1990. A constitutive model for partly saturated soils.

Géotechnique, 40(3): 405–430.

Al-Shayea, N., Abduljauwad, S., Bashire, R., Al-Ghamedy, H. and Asi, I. 2001.

Determination of parameters for a hyperbolic model of soils from the eastern

province of Saudi Arabia. Geotechnical Engineering, 149(4):1–14.

Anderson, J.U., Fadul, K.E. and O’Connor, G.A. 1973. Factors affecting the coefficient

of linear extensibility in vertisols. Soil Science Society of America Journal, 27:

296–299.

Azam, S. 2007. Study on the swelling behaviour of blended clay-sand soils. Geotechnical

and Geological Engineering, 25(3): 369–381.

Azam, S. and Ito, M. 2012. Coupled soil-atmosphere modeling for expansive Regina

clay. Journal of Environmental Informatics, 19(1): 20–29.

Bandyopadhyay, S.S. 1981. Prediction of swelling potential for natural soils. Journal of

Geotechnical Engineering, 107(5): 658–691.

214

Barden, L., Madedor, A.O. and Sides, G.R. 1969. Volume change characteristics of

unsaturated clay. Journal of the Soil Mechanics and Foundation Division, ASCE,

95: 33–52.

Bell, F.G. and Culshaw, M.G. 2001. Problem soils: a review from a British perspective.

In Proceedings of Problematic Soils Symposium (Jefferson, I., Murray, E.J.,

Faragher, E. and Fleming, P.R., eds.), Nottingham, pp. 1–35.

Benson, C.H. 2007. Modeling Unsaturated Flow and Atmospheric Interactions.

Theoretical and Numerical Unsaturated Soil Mechanics (Schanz, T. ed.), Springer

Proceedings in Physics 113, Berlin, 187–201.

Berardi, R. and Lancellotta, R. 1991. Stiffness of granular soil from field performance.

Geotechnique, 41(1): 149-157.

Biot, M.A. 1941. General theory for three-dimensional consolidation. Journal of Applied

Physics, 12(2): 155–164.

Bishop, A.W. 1959. The principle of effective stress. Lecture delivered in Oslo, Norway,

in 1955. Published in Teknisk Ukeblad, 106(39): 859–863.

Bishop, A.W. and Eldin, A.K.G. 1950. Undrained triaxial tests on saturated sands and

their significance in the general theory of shear strength. Géotechnique, 2(1): 13–

32.

Blatz, J.A. and Graham, J. 2003. Elastic-plastic modelling of unsaturated soil using

results from a new triaxial test with controlled suction. Géotechnique, 53(1): 113–

122.

Blight, G. E. 1965. A study of effective stresses for volume change. Moisture Equilibria

and Moisture Changes in Soils beneath Covered Areas (Aitchison, G.D., ed.).

Butterworths, Sydney, 259–269.

Bohnhoff, G.L., Ogorzalek, A.S., Benson, C.H., Shackelford, C.D. and Apiwantragoon,

P. 2009. Field data and water-balance predictions for a monolithic cover in a

215

semiarid climate. Journal of Geotechnical and Geoenvironmental Engineering,

135(3): 333–348.

Bolt, G. H. and Miller, R. D. 1958. Calculation of total and component potentials of

water in soils. American Geophysicist Union Transportaion, 39: 917–928.

Bolzon, G., Schrefler, B. A. and Zeinkiewicz, O.C. 1996. Elasto-plastic soil constitutive

laws generalised to partially saturated states. Géotechnique, 46: 279–289.

Bowles, J.E. 1987. Elastic foundation settlement on sand deposits. Journal of

Geotechnical Engineering, ASCE, 113(8): 846–860.

Brackley, I.J.A. 1971. Partial collapse in unsaturated expansive clay. In Proceedings of

the Fifth Regional Conference for Africa on Soil Mechanics Foundation and

Engineering, Luanda, Angola, 23–30.

Brackley, I.J.A. 1975. A model of unsaturated clay structure and its application to swell

behaviour. In Proceedings of the Sixth Regional Conference for Africa on Soil

Mechanics and Foundation Engineering, Durban, South Africa, pp. C26–C34.

Bratton, W.L. 1991. Parameters for predicting shrink/heave beneath slab-on-ground

foundations over expansive clays. PhD thsis, Texas Tech University, Lubbock, Tex.

Briaud, J.L., Zhang, X. and Moon, S. 2003. The shrink test–water content method for

shrink and swell prediction. Journal of Geotechnical and Geoenvironmental

Engineering, ASCE, 129(7): 590–600.

Buckingham, E. 1907. Studies of the movement of soil moisture. Bulletin 38. USDA

Bureau of Soils, Washington, DC.

Buckingham, E. 1914. On physically similar systems: illustrating the use of dimensional

analysis. Physical Review, 4: 345-376.

Burland, J. B. 1965. Some aspects of the mechanical behaviour of partly saturated soils.

Moisture Equilibria and Moisture Changes in the Soils beneath Covered Areas

216

(Aitchison, G.D., ed.), Butterworths, Sydney, pp. 270–278.

Butterfield, R. 1999. Dimensional analysis for geotechnical engineers. Geotechnique, 49:

357–366.

Buzzi, O. 2010. On the use of dimensional analysis to predict swelling strain.

Engineering Geology, 116: 149–156.

Buzzi, O., Giacomini, A. and Fityus, S. 2011. Towards a dimensional description of soil

swelling behaviour. Géotechnique, 61(3): 271–277.

Chao, K.C. 2007. Design principles for foundations on expansive soils. Ph.D. Thesis.

Colorado State University, Fort Collins, Colorado.

Chao, K.C., Overton, D.D., and Nelson, J.D. 2006. The effects of site conditions on the

predicted time rate of heave. In Proceedings of the Unsaturated Soils Conference.

Special Publication 147, ASCE, pp. 2086–2097.

Chen, F.H. 1975. Foundations on Expansive Soils. 1st edition. Elsevier, New York.

Ching, R.K.H. and Fredlund, D.G. 1984. A small Saskatchewan town copes with

swelling clay problems. In Proceedings of the Fifth International Conference on

Expansive Soils, Adelaide, Australia, pp. 306–310.

Chiu, C.F. and Ng, C.W.W. 2003. A state-dependent elastoplastic model for saturated

and unsaturated soils. Géotechnique, 53(9): 809–829.

Christophe, F., Sell, R. and Coatanéa, E. 2008. Conceptual design framework supported

by dimensional analysis and system modelling language. Estonian Journal of

Engineering, 14 (4): 303–316.

Clifton, A.W., Yoshida, R., Fredlund, D.G. and Chursinoff, R.W. 1984. Performance of

Darke Hall, Regina, Canada, constructed on highly swelling clay. In Proceedings of

the Fifth International Conference on Expansive Soils, Adelaide, Australia, pp.

197–201.

217

Clisby, M.B. 1963. Predicting the movement of clays. Annual Meeting of the Highway

Research Board, Mississippi State University, State College.

Cokca, E. 2002. Relationship between methylene blue value, initial soil suction and swell

percent of expansive soils. Turkish Journal of Engineering and Environmental

Sciences, 26: 521–529.

Coleman, J.D. 1962. Stress/strain relations for partly saturated soils. Géotechnique, 12(4):

348–350.

Costa, Y.D., Cintra, J.C. and Zornberg, J.G. 2003. Influence of matric suction on the

results of plate load tests performed on a lateritic soil deposit. Geotechnical Testing

Journal 26 (2): 219–227.

Cover, A.P. and Lytton, R.L. 2001. Estimating soil swelling behaviour using soil

classification properties. Expansive Clay Soils and Vegetative Inference on

Shallow Foundations. In Proceedings of Geo-Institute Shallow Foundation and Soil

Properties Committee Sessions at the ASCE 2001 Civil Engineering Conference,

Houston, Texas, pp. 44–63.

Croney, D., Coleman, J.D. and Black, W.P.W. 1958. Movement and distribution of water

in soil in relation to highway design and performance. National Research Council,

Highway Research Board, Special Rep, Washington, DC, 40: 226–252.

Cui, Y.J. and Sun, D.A. 2009. Constitutive modelling: from isothermal to non-isothermal

behaviour of unsaturated soils. In Proceedings of the Unsaturated Soils–Theoretical

and Numerical Advances in Unsaturated Soil Mechanics (Buzzi, O., Fityus, S.G.

and Sheng, D., eds.), Taylor & Francis Group, CRC Press, London, pp. 493–506.

Cui, Y.J., Delage, P. and Sultan, N. 1995. An elasto-plastic model for compacted soils. In

Proceedings of the First International Conference on Unsaturated Soils, UNSAT-

95, Paris, pp. 703–709.

Dakshanamurthy, V. and Fredlund, D.G. (1980). Moisture and air flow in an unsaturated

218

soil. In Proceedings of the Fourth International Conference on Expansive Soils,

Denver, Colorado, 1: 514–532.

Dakshanamurthy, V., Fredlund, D.G. and Rahardjo, H. 1984. Coupled three-dimensional

consolidation theory of unsaturated porous media. In Proceedings of the Fifth

International Conference on Expansive Soils, Adelaide, Australia, May 21–23, pp.

99–104.

Davis, A.G., and Poulos, H.G. 1968. The use of elastic theory for settlement prediction

under three dimensional conditions. Geotechnique, 18: 67–91.

Delage, P. and Graham, J. 1995. Mechanical behaviour of unsaturated soils:

understanding the behaviour of unsaturated soils requires reliable conceptual

models. In Proceedings of the First International Conference on Unsaturated Soils,

UNSAT-95, Paris, pp. 1223–1256.

Department of the Army USA. 1983. Technical Manual TM 5-818-7. Foundations in

Expansive Soils.

Dhowian, A.W., 1990. Simplified heave prediction model for expansive shale.

Geotechnical Testing Journal, 13(4): 323–333.

Dhowian, A.W., Erol, O. and Youssef, A. F. 1985. Evaluation of expansive soils and

foundation methodology in the kingdom of Saudi Arabia. Research Rep. No. AT-5-

88-3, CANCST.

Diewald, G.A. 2003. A modified soil suction heave prediction protocol: with new data

from Denver area expansive soil sites. M.Sc. Thesis. University of Colorado at

Denver, Denver, Colorado.

Donald, I. B. 1956. The mechanical properties of saturated and partly saturated soils with

special reference to negative pore water pressure. In Proceedings of the Second

Australia - New Zealand Conference Soil Mechanical Foundation Engineering,

Christchurch, New Zealand, pp. 200–205.

219

Donaldson, G.W.1969. The occurrence of problems of heave and the factors affecting its

nature. In Proceedings of the Second International Research and Engineering

Conference on Expansive Clay soils, Texas A & M Press.

Driscoll, R. and Crilly, M. 2000. Subsidence damage to domestic buildings. Lessons

learned and questions asked. Building Research Establishment, London.

Duncan, J.M. and Chang C.Y. 1970. Nonlinear analysis of stress and strain in soils.

Journal of the Soil Mechanics and Foundations Division, ASCE, 96(SM5): 1926–

1953.

Dye, H.B. 2008. Moisture movement through expansive soil and impact on performance

of residential structures. Ph.D. Thesis. Arizona State University, Tempe, Arizona.

Erol, A.A. and Dhowian, A. 1990. Swell behavior of arid climate shales from Saudi

Arabia. The Quarterly Journal Engineering Geology, London, 23: 243–254.

Fayer, M. 2000. UNSAT-H Version 3.0: Unsaturated Soil Water and Heat Flow Model –

Theory, User Manual, and Examples. Report No. PNNL-13249. Pacific Northwest

National Laboratory, Richland, WA.

Fityus, S.G., Smith, D.W. and Allman, M.A. 2004. An expansive soil test site near

Newcastle. Journal of Geotechnical and Geoenvironmental Engineering, ASCE,

130(7): 686–695.

Fredlund, D.G. 1979. Appropriate concepts and technology for unsaturated soils. Second

Canadian Geotechnical Colloquium, Canadian Geotechnical Journal, 16(1): 121–

139.

Fredlund, D.G. 1983. Prediction of ground movements in swelling clays, paper presented

to the Thirty-First Annual ASCE Soil Mechanics and Foundation Engineering

Conference, Earle Brown Center, University of Minnesota, Minneapolis, MN.

Fredlund, D.G. 1997. An introduction to unsaturated soil mechanics. ASCE Geotechnical

Engineering Division, GeoLogan Conference, Special Geotechnical Publication

220

(68): 1–37, Logan, Utah.

Fredlund, D.G. and Hasan, J. 1979. One-dimensional consolidation theory, unsaturated

soils. Canadian Geotechnical Journal, 16(3): 521–531.

Fredlund, D.G. and Houston, S.L. 2013. Interpretation of soil-water characteristic curves

when volume change occurs as soil suction is changed. In Proceedings of the First

Pan-Am Conference on Unsaturated Soils, UNSAT 2013, "Advances in

Unsaturated Soils", Editor: Bernardo Caicedo, February 20-22, pp. 15–31.

Fredlund, D.G. and Morgenstern, N.R. 1976. Constitutive relations for volume change in

unsaturated soils. Canadian Geotechnical Journal, 13: 261–276.

Fredlund, D.G. and Morgenstern, N.R. 1977. Stress state variables for unsaturated soils.

Journal of Geotechnical Engineering Division, ASCE, 103(GT5): 447–466.

Fredlund, D.G. and Rahardjo, H. 1993. Soil Mechanics for Unsaturated Soil. John Wiley

& Son, Inc., New York.

Fredlund, D.G. and Vu, H.Q. 2001. Prediction of volume change in an expansive soil as a

result of vegetation and environmental changes. In Proceedings of the ASCE

Conference on Expansive Clay Soils and Vegetative Influences on Shallow

Foundations, Geotechnical Special Publication, Houston, Texas, Oct. 10-13, 115,

pp. 24–43.

Fredlund, D.G. and Xing, A. 1994. Equations for the soil-water characteristic curve.

Canadian Geotechnical Journal, 31(4): 521–532.

Fredlund, D.G., Hasan, J.U. and Filson, H.L. 1980. The prediction of total heave. In

Proceedings of the Fourth International Conference on Expansive Soils, Denver,

CO, pp. 1–17.

Fredlund, D.G., Rahardjo, H. and Fredlund, M.D. 2012. Unsaturated Soils Mechanics in

Engineering Practice. Wiley, New York.

221

Freeze, R.A. and Cherry, J.A. 1979. Groundwater. Prentice-Hall Inc., Englewood Cliffs,

New Jersey.

Fung, Y.C. 1977. A First Course in Continuum Mechanics. 2nd edition. Prentice-Hall:

Englewood Cliffs, New Jersey.

Gallipoli, D., Gens, A., Sharma, R. and Vaunat, J. 2003. An elasto-plastic model for

unsaturated soil incorporating the effects of suction and degree of saturation on

mechanical behaviour. Géotechnique, 53(1): 123–135.

Geiser, F., Laloui, L. and Vulliet, L. 2000. Modelling the behaviour of unsaturated silt.

Experimental Evidence and Theoretical Approaches in Unsaturated Soils. In

Proceedings of an International Workshop, Trento, pp. 155–175.

Gens, A. 1996. Constitutive modelling: application to compacted soils. In Proceedings of

the First International Conference on Unsaturated Soils (Alonso, E.E. and Delage,

P., eds.), Balkema, Rotterdam, pp. 1179–1200.

Gens, A. 2009. Some issues in constitutive modelling of unsaturated soils. In proceedings

of the Unsaturated Soils – Theoretical and Numerical Advances in Unsaturated Soil

Mechanics (Buzzi, O., Fityus, S.G. and Sheng, D., eds.), CRC Press, pp. 613–626.

Gens, A. 2010. Soil-environmental interactions in geotechnical engineering.

Géotechnique, 60: 3–74.

Gens, A. and Alonso, E.E. 1992. A framework for the behaviour of unsaturated

expansive clays. Canadian Geotechnical Journal, 29:1013–1031.

Gens, A., Sánchez, M. and Sheng, D. 2006. On constitutive modelling of unsaturated

soils. Acta Geotechnica, 1: 137–147.

Gens, A., Alonso, E.E., Suriol, J. and Lloret, A. 1995. Effect of structure on the

volumetric behaviour of a compacted soil. In Proceedings of the First International

Conference on Unsaturated Soils, UNSAT-95, Paris, pp. 83–88.

222

Geo-Slope. 2005. GEO-STUDIO VADOSE/W Software Package for Seepage Analysis.

Version 6.16. Calgary, Alberta.

Geo-Slope. 2007. Vadose Zone Modeling with VADOSE/W 2007: An Engineering

Methodology. 3rd Edition. Geo-Slope, Calgary, Alberta.

Grim, R.E. 1968. Clay Mineralogy. McGraw-Hill, New York.

Hamberg, D.J. 1985. A simplified method for predicting heave in expansive soils. M.Sc.

Thesis. Colorado State University, Fort Collins, CO.

Hilf, J.W. 1956. An investigation of pore-water pressure in compacted cohesive soils.

Ph.D. Thesis. Technical Memorandum No. 654, U.S. Department of the Interior,

Bureau of Reclamation, Design and Construction Division, Denver, CO.

Holland, J.E. and Cameron, D.A. 1981. Seasonal heave of clay soils. Institution of

Engineers of Australia, Civil Engineering Transactions, CE23(1): 55–67.

Holtz, W.G. and Gibbs, H.J. 1954. Engineering properties of expansive clays. In

Proceedings of ASCE, 80(516): 1–28.

Holtz, W.G. and Gibbs, H.J. 1956. Engineering properties of expansive clays.

Transactions ASCE, 121: 641–663.

Inci, G., Yesiller, N. and Kagawa, T. 2003. Experimental investigation of dynamic

response of compacted clayey soils. Geotechnical Testing Journal, 26 (2): 125–141.

Ito, M. and Hu, Y. 2011. Prediction of the behaviour of expansive soils. In Proceedings

of the 2011 Pan-Am CGS Conference, Toronto, Ontario.

Janbu, N. 1963. Soil compressibility as determined by oedometer and triaxial test. In

Proceedings of the European Conference on Soil Mechanics and Foundation

Engineering, Wiesbaden, pp. 19–25.

Jennings, J.E.B. 1961. A revised effective stress law for use in the prediction of the

behavior of unsaturated soils. In Proceedings of the Conference on Pore Pressure

223

and Suction in Soils, Butterworths, London, UK, pp. 26–30.

Jennings, J.E.B. 1969. The prediction of amount and rate of heave likely to be

experienced in engineering construction on expansive soils. In Proceedings of the

Second International Conference on Expansive Clay Soils, Texas A & M

University, College Station, TX, pp. 99–109.

Jennings, J.E.B. and Burland, J.B. 1962. Limitations to the use of effective stresses in

partly saturated soils, Géotechnique, 12(2): 125–144.

Jennings, J.E.B. and Knight, K. 1957. The prediction of total heave from double

oedometer test. In Proceedings of Symposium on Expansive Clays, South African

Institution of Civil Engineers, Johannesburg, pp. 13–19.

Johnson, L.D. 1973. Influence of suction on heave of expansive soils. Miscellaneous

Paper S-73-17, US Army Engnrs Waterways Exp. Station, Vicksburg, MS.

Johnson, L.D. 1977. Evaluation of laboratory suction tests for prediction of heave in

foundation soils. Rep. WES-TR-S-77-7, August. US Army Engnrs, Waterways

Exp. Station, Vicksburg, MS.

Johnson, L.D. 1978. Predicting potential heave and heave with time in swelling

foundation soils. US Army Engineer Waterways Experiment Station, CE,

Vicksburg, Miss. Technical report S-78-7.

Johnson, L.D. and Snethen, D.R. 1978. Prediction of potential heave of swelling soils.

Geotechnical Testing Journal, 1(3): 117–124.

Jommi, C. 2000. Remarks on the constitutive modeling of unsaturated soils.

Experimental Evidence and Theoretical Approaches in Unsaturted Soils (Tarantino,

A. and Mancuso, C., eds.), Balkema, Rotterdam, pp. 139–153.

Jones, D.E. and Holtz, W.G. 1973. Expansive soils - the hidden disaster, Civil

Engineering, 43(8): 49–51.

224

Jones, L.D. and Jefferson, I. 2012. Expansive soils. ICE Manual of Geotechnical

Engineering (Burland, J., ed.), Volume 1, Geotechnical Engineering Principles,

Problematic Soils and Site Investigation. London, UK, ICE Publishing.

Jotisankasa, A., 2005. Collapse behaviour of a compacted silty clay, Ph.D. Thesis.

Imperial College, London, UK.

Justo, J.L. and Saetersdal, R. 1981. Design parameters for special soil conditions. General

Report. In Proceedings of the Seventh European Conference of Mechanics,

Brighton, UK, pp. 127–158.

Kalantari, B. 2012. Engineering significant of swelling soils. Research Journal of Applied

Sciences, Engineering and Technology, 4(17): 2874–2878

Karube, D. 1988. New concept of effective stress in unsaturated soil and its proving test.

In Proceedings of the Advanced Triaxial Testing of Soil and Rock (Donaghe, R.T.,

Chaney, R.C. and Silver, M.L., eds.), ASTM STP 977, pp. 539–552.

Kato, S., Matsuoka, H. and Sun, D. A. 1995. A constitutive model for unsaturated soil

based on extended SMP. In Proceedings of the First International Conference on

Unsaturated Soils, UNSAT-95, Paris, pp. 739–744.

Khalili, N. and Khabbaz, M.H. 1998. A unique relationship fo chi for the determination

of the shear strength of unsaturated soils. Geotechnique, 48 (5): 681–687.

Khalili, N., Geiser F. and Blight G.E. 2004. Effective stress in unsaturated soils: review

with new evidence, International Journal of Geomechanics, 4(2): 115–126.

Kohgo, Y. 2003. Review of constitutive models for unsaturated soils and initial-boundary

value analyses. In Proceedings of the Second Asian Conference on Unsaturated

Soils, UNSAT-ASIA 2003, Osaka, pp. 21–40.

Kohgo, Y., Nakano, M. and Miyazaki, T. 1993. Theoretical aspects of constitutive

modelling for unsaturated soils. Soils and Foundations, 33(4): 49–63.

225

Krahn, J. and Fredlund, D.G. 1972. On total matric and osmotic suction. Journal of Soil

Science Journal, 114(5): 339–348.

Krohn, J.P. and Slosson, J. 1980. Assessment of expansive soils in the United States. In

Proceedings of the Fourth International Conference on Expansive Soils, Denver,

CO, pp. 596–608.

Lade, P.V. 1988. Model and parameters for the elastic behavior of soils. In Proceedings

of the International Conference on Numerical Methods in Geomechanics,

Innsbruck, Austria (Swoboda, G., ed.). Balkema, Rotterdam, the Netherlands.

Lade, P.V. and Nelson, R.B. 1987. Modeling the elastic behavior of granular materials.

International Journal for Numerical and Analytical Methods in Geomechanics,

11(5): 521–542.

Lambe, T.W. and Whitman, R.V. 1959. The role of effective stress in the behavior of

expansive soils. Quarterly of the Colorado School of Mines, 54(4): 33–66.

Lambe, T.W. and Whitman, R.V. 1979. Soil Mechanics. Wiley, New York.

Lancelotta, R. 1995. Geotechnical Engineering. Balkema, Rotterdam, the Netherlands.

Laughton, A.S. 1955. The compaction of ocean sediments. Ph.D. Thesis. University of

Cambridge, England.

Lee, J., Eun, J., Prezzi, M. and Salgado, R. 2008. Strain influence diagrams for settlement

estimation of both isolated and multiple footings in sand. Journal of Geotechnical

and Geoenvironmental Engineering, ASCE, 134(4): 417–427.

Leong, E.C. and Rahardjo, H. 1997. Permeability functions for unsaturated soils. Journal

of Geotechnical and Geoenvironmental Engineering, ASCE, 123(12): 1118–1126.

Lian, G., Thornton, C. and Adams, M. J. 1993. A theoretical study of the liquid bridge

forces between two rigid spherical bodies. Journal of Colloid Interface Science,

161: 138–148.

226

Lloret, A. and Alonso, E.E. 1980. Consolidation of unsaturated soils including swelling

and collapse behaviour. Géotechnique, 30(4): 449–477.

Lloret, A., Gens, A., Battle, F. and Alonso, E.E. 1987. Flow and deformation analysis of

partially saturated soils. In Proceedings of the 9th European Conference on Soil

Mechanics and Foundation Engineering, Dublin, (Hanarahan, Orr, Widdis, eds.),

Balkema, Rotterdam, Netherlands, pp. 565–568.

Lu, L. 2010. A simple technique for estimating the 1-D heave of natural expansive soils.

M.Sc. Thesis. University of Ottawa, Ottawa, Ontario.

Lu, N. and Kaya, M. 2014. A power law for elastic moduli of unsaturated soil. Journal of

Geotechnical and Geoenvironmental Engineering,140(1), 46–56.

Lu, N. and Likos, W.J. 2006. Suction stress characteristic curve for unsaturated soils.

Journal of Geotechnical and Geoenvironmental Engineering, 132(2): 131–142.

Lytton, R.L. 1977. The characterization of expansive soils in engineering. Presented at

Symposium on Water Movement and Equilibria in Swelling Soils, American

Geophysical Union, San Francisco, Calif.

Lytton, R.L. 1994. Prediction of movement in expansive clays. In Proceedings of the

Settlement ’94 Conference, College Station, Texas, Geotechnical Special

Publication, ASCE, New York, 2(40): 1827–1845.

Lytton, R.L. and Kher, R.K. 1970. Prediction of moisture movement in expansive clays.

Issue 118, Part 3 of Research report, Center for Highway Research, University of

Texas at Austin.

Lytton, R.L., Aubeny, C. and Bulut, R. 2004. Design procedure for pavements on

expansive soils. Report No. FHWA/TX-05/0-4518-1. Texas Department of

Transportation, Austin, Texas.

Matyas, E.L. and Radhakrishna, H.S. 1968. Volume change characteristics of partially

saturated soils. Géotechnique, 18(4): 432–448.

227

Mayne, P.W. and Poulos, H.G. 1999. Approximate displacement influence factors for

elastic shallow foundations. Journal of Geotechnical and Geoenvironmental

Engineering, ASCE, 125(6): 453–460.

McCormack, D.E. and Wilding, L.P. 1975. Soil properties influencing swelling in

Canfield and Geeburg soils. Soil Science Society of America Journal, 39(3): 496–

502.

McKeen R.G. 1980. Field study of airport pavements on expansive clay. In Proceedings

of the Fourth International Conference on Expansive Soils, Denver, CO, pp. 242–

261.

McKeen, R.G. 1981. Design of airport pavements for expansive soils. Federal Aviation

Agency, US Department of Transportation, Washington, DC.

McKeen, R.G. 1992. A model for predicting expansive soil behavior. In Proceedings of

the Seventh International Conference on Expansive Soils, Dallas, TX.

McKeen, R.G. and Johnson, L.D. 1990. Climate-controlled soil design parameters for

mat foundations. Journal of Geotechnical Engineering, 116(7): 1073–1094.

McKeen, R.G. and Nielsen, J.P. 1978. Characterization of expansive soils for airport

pavement design. US Department of Transportation, Federal Aviation

Administration, FFA Rep. FAA-RD-78-59.

Miao, L., Liu, S. and Lai, Y. 2002. Research of soil-water characteristics and shear

strength features of Nanyang expansive soil. Engineering Geology, 65(4): 261–267.

Miao, L., Houston, S.L., Cui, Y. and Yuan, J. 2007. Relationship between soil structure

mechanical behavior for an expansive unsaturated clay. Canadian Geotechnical

Journal 44(2): 126–137.

Miller, D.J. and Nelson, J.D. 2006. Osmotic suction in unsaturated soil mechanics. In

Proceedings of the Fourth International Conference on Unsaturated Soils, Carefree,

AZ, Geotechnical Special Publication No. 147, 1382–1393.

228

Mitchell, J.K. 1993. Fundamentals of Soil Behaviour. John Wiley & Sons, Inc., London.

Mitchell, J.K. and Soga, K. 2005. Fundamentals of Soil Behavior. 3rd Edition. John

Wiley & Sons, Inc., New Jersey.

Mitchell, P.W. 1979. The structural analysis of footings on expansive soil. Kenneth W.G.

Smith and Associates Research Report No. 1. 1st Edition. Newton, South Australia.

Mitchell, P.W. and Avalle, D.L. 1984. A technique to predict expansive soils movement.

In Proceedings of the Fifth International Conference on Expansive Soils, Adelaide,

Australia, pp. 124–130.

Modaressi, A. and Abou-Bekr, N. 1994. Constitutive model for unsaturated soils:

validation on a silty material. In Proceedings of Numerical Methods in

Geotechnical Engineering, Manchester, pp. 91–96.

NAVFAC, 1971. Soil mechanics foundation and earth structures, Design Manual,

NAVFACDM-7, Naval Facilities Engineering Command, Department of the Navy,

Bureau of Yards and Docks, Washington, DC.

Nayak, N.V. and Christensen, R.W. 1971. Swelling characteristics of compacted,

expansive soil. Clays and Clay Minerals, 19(4): 251–261.

Nelson, J.D. and Miller, D.J. 1992. Expansive Soils: Problems and Practice in Foundation

and Pavement Engineering. Wiley, New York.

Nelson, J.D., Chao, K.C. and Overton, D.D. 2007. Definition of expansion potential for

expansive soils. In Proceedings of the Third Asian Conference on Unsaturated

Soils.

Nelson, J.D., Overton, D.D. and Durkee, D.B. 2001. Depth of wetting and the active

zone. In Proceedings of Expansive Clay Soils and Vegetative Influence on Shallow

Foundations, Proceedings of the Geo-Institute Foundations and Soil Properties

Committee Sessions, Geotechnical Special Publication No. 115, ASCE, Houston,

TX, 132–157.

229

Nelson, J.D., Reichler, D.K. and Cumbers, J.M. 2006. Parameters for heave prediction by

oedometer tests. In Proceedings of the Fourth International Conference on

Unsaturated Soils, Carefree, Arizona, ASCE, Geotechnical Special Publication,

(147): 951–961.

Nelson, J.D., Chao, K.C., Overton, D.D. and Schaut, R.W. 2011. Calculation of heave of

deep pier foundations. In Proceedings of the Fifth Asia-Pacific Conference on

Unsaturated Soils: Theory and Practice 2011, Kasetsart University, Thailand, 59–

74.

Nelson, J.D., Chao, K.C., Overton, D.D. and Schaut, R.W. 2012. Calculation of heave of

deep pier foundations. SEAGS and AGSSEA Geotechnical Engineering Journal,

43(1): 12–25.

Newland, P.L. 1965. The behavior of soils in terms of two kinds of effective stress. In

Proceedings of International Research and Engineering Conference on Expansive

Clay Soils, College Station, Texas, 78–92.

Ng, C.W.W., Zhan, L.T., Bao, C.G., Fredlund, D.G. and Gong, B.W. 2003. Performance

of an unsaturated expansive soil slope subjected to artificial rainfall infiltration.

Géotechnique 53(2): 143–157.

Noble, C.A. 1966. Swelling measurements and prediction of heave for a lacustrine clay.

Canadian Geotechnical Journal, 3(1): 32–41.

Nur, A. and Byerlee, J. 1971. Exact effective stress law for elastic deformation of rocks

with fluids. Journal of Geophysical Research; 76(26): 6414–6419.

Nuth, M. and Laloui, L. 2008. Effective stress concept in unsaturated soils: clarification

and validation of an unified framework. International Journal of Numerical and

Analytical Methods in Geomechanics, 32: 771–801.

Oh, W.T., Vanapalli, S.K. and Puppala, A.J. 2009. Semi-empirical model for the

prediction of modulus of elasticity for unsaturated soils. Canadian Geotechnical

230

Journal, 46(8): 903–914.

Osman, N.Y., McManus, K. and Ng, A.W.M. 2005. Management and analysis of data for

damage of light structures on expansive soils In Victoria, Australia. In Proceedings

of the First International Conference on Structural Condition Assessment,

Monitoring and Improvement, Perth, Australia, CI-Premier, Singapore, pp. 283–

290.

Overton, D.D., Chao, K.C. and Nelson, J.D. 2006. Time rate of heave prediction for

expansive soils. In Proceedings of GeoCongress 2006, ASCE, Atlanta.

O’Neil, M.W. and Ghazzally, O.I. 1977. Swell potential related to building performance.

Journal of the Geotechnical Engineering Division, ASCE, 103(12): 1363–1379.

Palmer, A.C. 2008. Dimensional Analysis and Intelligent Experimentation. Singapore:

World Scientific.

Pereira, J.H.F. 1996. Numerical analysis of the mechanical behavior of collapsing earth

dams during first reservoir filling. Ph.D. Thesis. University of Saskatchewan,

Saskatoon, Sask.

Pufahl, D.E. and Lytton, R.L. 1992. Temperature and suction profiles beneath highway

pavements: computed and measured. Transportation Research Record 1307,

Transportation Research Board, Washington, D.C., 268–276.

Puppala, A.J. and Cerato, A. 2009. Heave distress problems in chemically-treated sulfate-

laden materials. Geo-Strata, 10(2): 28–30, 32.

Rahardjo, H., Melinda, F., Leong, E.C. and Rezaur, R.B. 2011. Stiffness of a compacted

residual soil. Engineering Geology, 120: 60–67.

Rampino, C., Mancuso, C. and Vinale, F. 2000. Experimental behaviour and modeling of

an unsaturated compacted soil. Canadian Geotechnical Journal, 37(4): 748–763.

Ranganatham, B.V., and Satyanarayan, B. 1965. A rational method of predicting swelling

231

potential for compacted expansive clays. In Proceedings of the Sixth International

Conference on Soil Mechanics and Foundation Engineering, Montreal, QC, pp. 92–

96.

Rao, A.S., Phanikumar, B.R. and Sharma, R.S. 2004. Prediction of swelling

characteristics of remoulded and compacted expansive soils using free swell index.

Quarterly Journal of Engineering Geology and Hydrogeology, 37: 217–226.

Rao, B.H., Venkataramana, K. and Singh, D.N. 2011. Studies on the determination of

swelling properties of soils from suction measurements. Canadian Geotechnical

Journal, 48: 375–387.

Rassam, D.W. and Williams, D.J. 1999. A relationship describing the shear strength of

unsaturated soils. Canadian Geotechnical Journal, 36(2): 363–368.

Rendulic, L. 1936. Relation between void ratio and effective principal stress for a

remolded silty clay. In Proceedings of the First International Conference on Soil

Mechanics and Foundation Engineering, Cambridge, MA, pp. 48-51.

Richards, B.G. 1965. Measurement of the free energy of soil moisture by the

psychrometric technique using thermistors, Moisture Equilibria and Moisture

Changes in Soils Beneath Covered Areas (Aitchison, G. D., ed.), A Symposium in

Print, Butterworths, Sydney, pp. 39–46.

Richards, B.G. 1967. Moisture flow and equilibria in unsaturated soils for shallow

foundations. Special Technical Publication No. 417, ASTM, Philadelphia, pp. 4–

34.

Richards, B.G. 1974. Behaviour of unsaturated soils (Lee, I.K., ed.), Soil Mechanics–

New Horizons, American Elsevier, New York, pp. 112–157.

Richards, L.A. 1931. Capillary conduction of liquids through porous mediums. Physics,

1: 318–333.

Ruwaih, I.A. 1987. Experiences with expansive soils in Saudi Arabia. In Proceedings of

232

the Sixth International Conference on Expansive Soils, New Delhi, India, pp. 317–

322.

Salas, J.A.J. and Serratosa, J.M. 1957. Foundations on swelling clays. In Proceedings of

the Fourth International Conference on Soil Mechanics and Foundation

Engineering, London, England, pp. 424–428.

Schafer, W.M., Singer, M.J. 1976. Influence of physical and mineralogical properties on

swelling of soils in Yolo County, California. Soil Science Society of America

Journal, 40: 557–562.

Schmertmann, J.H. 1970. Static cone to compute static settlement over sand. Journal of

the Soil Mechanics and Foundations Division, ASCE, 96(3): 1011–1043.

Schmertmann, J.H., Hartmann, J.P. and Brown, P.R. 1978. Improved strain influence

factor diagrams. Journal of the Geotechnical Engineering Division, ASCE, 104(8):

1131–1135.

Schneider, G.L., and Poor, A.R. 1974. The prediction of soil heaves and swell pressures

developed by expansive clay. Construction Research Centre, CRC Publications,

Tex. Research Report TR-9-74.

Seed, H.B., Woodward, R.J. and Lundgren, R. 1962. Prediction of swelling potential for

compacted clays. Journal of the Soil Mechanics Foundation Division, ASCE, 88

(SM4): 57–59.

Sheng, D. 2011. Constitutive modelling of unsaturated soils: discussion of fundamental

principles. In Proceedings of the Unsaturated Soils (Alonso, E. E. and Gens, A.,

eds.), Taylor & Francis Group, CRC Press, London, pp. 91–112.

Sheng, D. and Fredlund, D.G. 2008. Elastoplastic modelling of unsaturated soils: an

overview. Keynote Lecture. In Proceedings of the 12th International Conference of

International Association for Computer Methods and Advances in Geomechanics

(IACMAG), Goa, India, pp. 2084-2105.

233

Sheng, D., Fredlund, D.G. and Gens, A. 2008. A new modelling approach for unsaturated

soils using independent stress variables. Canadian Geotechnical Journal, 45: 511–

534.

Shuai, F. 1996. Simulation of swelling pressure measurements on expansive soils. Ph.D.

Thesis. University of Saskatchewan, Saskatoon, Sask.

Singhal, S. 2010. Expansive soil behavior: property measurement techniques and heave

prediction methods. Ph.D. Thesis. Arizona State University, Tempe, Arizona.

Simunek, J., Seina, M. and van Genuchten, M. 1999. The HYDRUS-2D Software

Package for Simulating Two-dimensional Movement of Water, Heat, and Multiple

Solutes in Variably Saturated Media. Version 2.0. IGWMC-TPS-53, International

Ground Water Modeling Center, Colorado School of Mines, Golden, Colorado.

Skempton, A.W. 1961. Effective stress in soils, concrete and rocks. In Proceedings of the

Conference on Pore Pressure and Suction in Soils, Butterworths, London,UK, pp.

4–16.

Smith, A.W. 1973. Method for determining the potential vertical rise, PVR. Texas Test

Method Tex-126–E. In Proceedings of Workshop on Expansive Clays and Shales in

Highway Design and Construction, University of Wyoming, Laramie, WY, 1: 189–

205.

Snethen, D.R. 1980. Characterization of expansive soils using soil suction data. In

Proceedings of the Fourth International Conference on Expansive Soils. Denver,

Colorado, pp. 54–75.

Snethen, D.R. 1986. Expansive soils: where are we? Ground failure, No. 3. National

Research Council Communication on Ground Failure Hazards, National Research

Council, Washington, DC, pp. 12–16.

Snethen, D.R. and Huang, G. 1992. Evaluation of soil suction-heave prediction methods.

In Proceedings of Seventh International Conference on Expansive Soils. Dallas,

TX.

234

SoilVision Systems Ltd. 2007. SVFlux User’s Manual. SoilVision Systems Ltd.,

Saskatoon, SK.

Stavridakis, E.I. 2006. Assessment of anisotropic behaviour of swelling soils on ground

and construction work. Published in Expansive Soils: Recent Advances in

Characterization and Treatment (Al-Rawas, A.A. and Goosen, M.F.A., eds.).

Taylor and Francis, London, UK, Chapter 25.

Steinberg, M. 1998. Geomembranes and the control of expansive soils in construction.

McGraw-Hill, New York.

Sullivan, R.A. and McClelland, B. 1969. Predicting heave of buildings on unsaturated

clay. In Proceedings of the Second International Research and Engineering

Conference on Expansive Clay soils, Texas A & M University Press. College

Station, TX, pp. 404–420.

Taboada, M.A. 2003. Soil shrinkage characteristics in swelling soils. Lecture given at the

College on Soil Physics Trieste, 21 March 2003, Buenos Aires, Argentina.

Tamagnini, R. 2004. An extended cam-clay model for unsaturated soils with hydraulic

hysteresis. Géotechnique, 54(3): 223–228.

Tang, A., Cui, Y., Trinh, V., Szerman, Y. and Marchadier, G. 2009. Analysis of the

railway heave induced by soil swelling at a site in southern France. Engineering

Geology, 106: 68–77.

Tarantino, A., Mongiovi, L. and Bosco, G. 2000. An experimental investigation on the

independent isotropic stress variables for unsaturated soils, Géotechnique, 50(3):

275–282.

Terzaghi, K. 1925. Principles of soil mechanics: I- Phenomena of cohesion of clays.

Engineering News-Record, 95(19): 742–746.

Terzaghi, K. 1926. The mechanics of adsorption and swelling of gels. In Proceedings of

the Fourht Colloid Symposium Monograph, Chemical Catalog Co., New York, pp.

235

58–78.

Terzaghi, K. 1931. The influence of elasticity and permeability on the swelling of two-

phase systems. Colloid Chemistry (Alexander, J., ed.), New York, Chemical

Catalog Co., pp. 65–88.

Terzaghi, K. 1936. The shear resistance of saturated soils. In Proceedings of the First

International Conference on Soil Mechanics and Foundation Engineering,

Cambridge, MA, 1: 54–56.

Terzaghi, K. 1943. Theoretical Soil Mechanics. Wiley, New York.

Terzaghi, K., Peck, R.B. and Mesri, G. 1996. Soil Mechanics in Engineering Practice. 3rd

Ed. John Wiley & Sons, New York.

Thu, T.M., Rahardjo, H. and Leong, E.C. 2007. Elastoplastic model for unsaturated soil

with the incorporation of soil-water characteristic curve. Canadian Geotechnical

Journal, 44: 67–77.

Tourtelot, H.A. 1973. Geologic origin and distribution of swelling clays. In Proceedings

of Workshop on Expansive Clay and Shale in Highway Design and Construction, I.

Türköz, M. and Tosun, H. 2011. The use of methylene blue test for predicting swell

parameters of natural clay soils. Scientific Research and Essays, 6(8): 1780–1792.

Unsaturated Soils Group. 1996. SoilCover User’s Manual. Version 3.0. Unsaturated Soils

Group, University of Saskatchewan, Saskatoon, SK.

Vanapalli, S.K. 1994. Simple test procedures and their inter-pretation in evaluating the

shear strength of unsaturated soils. Ph.D. thesis. Department of Civil Engineering,

University of Saskatchewan, Saskatoon, Sask.

Vanapalli, S.K. and Lu, L. 2012. A state-of-the art review of 1-D heave prediction

methods for expansive soils. International Journal of Geotechnical Engineering,

6(1): 15–41.

236

Vanapalli, S.K. and Mohamed, F.M.O. 2007. Bearing capacity of model footings in

unsaturated soils. In Proceedings of the Experimental Unsaturated Soil Mechanics.

Springer-Verlag, Berlin Heidelberg, Germany, pp. 483–493.

Vanapalli, S.K. and Mohamed, F.M.O. 2013. Bearing capacity and settlement of footings

in unsaturated sands. International Journal of GEOMATE, 5(1): 595–604.

Vanapalli, S.K. and Oh, W.T. 2010. A model for predicting the modulus of elasticity of

unsaturated soils using the soil-water characteristic curve. International Journal of

Geotechnical Engineering, 4(4): 425–433.

Vanapalli, S.K., Lu, L. and Oh, W.T. 2010. Estimation of swelling pressure and 1-D

heave in expansive soils. In Proceedings of the Fifth International Conference on

Unsaturated Soils, Barcelona, Spain, pp. 1201–1207.

Vanapalli, S.K., Fredlund, D.G., Pufahl, D.E. and Clifton, A.W. 1996. Model for the

prediction of shear strength with respect to soil suction. Canadian Geotechnical

Journal, 33(3): 379–392.

van Genuchten, M.T. 1980. A closed-form equation for predicting the hydraulic

conductivity of unsaturated soils. Journal of Soil Science Society of America, 44:

892–898.

Vijayvergiya, V.N. and Ghazzally, O.I. 1973. Prediction of swelling potential of natural

clays. In Proceedings of the Third International Conference on Expansive Clays,

Haifa, pp. 227–234.

Vu, H.Q. 2002. Uncoupled and coupled solutions of volume change problems in

expansive soils. Ph.D. thesis. Department of Civil Engineering, University of

Saskatchewan, Saskatoon, Sask.

Vu, H.Q. and Fredlund, D.G. 2004. The prediction of one-, two-, and three-dimensional

heave in expansive soils. Canadian Geotechnical Journal, 41(4): 713–737.

Vu, H.Q. and Fredlund, D.G. 2006. Challenges to modelling heave in expansive soils.

237

Canadian Geotechnical Journal, 43(12): 1249–1272.

Vu, H.Q., Hu, Y. and Fredlund, D.G. 2007. Analysis of soil suction changes in expansive

Regina clay. In Proceedings of the 60th Canadian Geotechnical Conference,

Ottawa, Ontario, pp. 1069–1076.

Wanyan,Y., Manosuthkij, T., Abdallah, I., Nazarian, S. and Puppala, A. J. 2008. Expert

system design guide for lower classification roads over high pi clays. Research

Report 0-5430-2, TXDOT Project Number 0-5430. Performed in Cooperation with

the Texas Department of Transportation & Federal Highway Administration.

Weston, D.J. 1980. Expansive roadbed treatment for Southern Africa. In Proceedings of

the Fourth International Conference on Expansive Soils, Denver, CO., ASCE, pp.

339–360.

Wheeler, S.J. and Karube, D. 1996. Constitutive modelling. In Proceedings of the First

International Conference on Unsaturated Soils (Alonso, E.E. and Delage, P., eds.),

Balkema, Rotterdam, pp. 1323–1356.

Wheeler, S.J. and Sivakumar, V. 1995. An elastoplastic critical state framework for

unsaturated soil. Géotechnique, 45(1): 35–53.

Wheeler, S.J., Sharma, R.J. and Buisson, M.S.R. 2003. Coupling of hydraulic hysteresis

and stress–strain behaviour in unsaturated soils. Géotechnique, 53(1): 41–54.

Wilson, G.W. 1990. Soil evaporative fluxes for geotechnical engineering problems. Ph.D.

Thesis. University of Saskatchewan, Saskatoon, Sask.

Wong, T.T., Fredlund, D.G. and Krahn, J. 1998. A numerical study of coupled

consolidation in unsaturated soils. Canadian Geotechnical Journal, 35(6): 926–937.

Wray, W.K. 1984. The principle of soil suction and its geotechnical engineering

applications. In Proceedings of the Fifth International Conference on Expansive

Soils, Adelaide, Australia, pp. 114–118.

238

Wray W.K., 1989 . Mitigation of damage to structures supported on expansive soils, Vol.

1, Final Report, National Science Foundation, Washington, DC.

Wray, W.K. 1997. Using soil suction to estimate differential soil shrink or heave. In

Unsaturated Soil Engineering Practice. Geotechnical Special Publication no. 68,

ASCE, pp. 66–87.

Wray, W.K. 1998. Mass transfer in unsaturated soils: a review of theory and practice.

State-of-the-Art Report. In Proceedings of the Second International Conference on

Unsaturated Soils, Beijing, II: 99–155.

Wray, W.K., El-Garhy, B.M. and Youssef, A.A. 2005. Three-dimensional model for

moisture and volume changes prediction in expansive soils. Journal of Geotechnical

and Geoenvironmental Engineering, 131(3): 311–324.

Yang, S.R., Huang, W.H. and Tai, Y.T. 2005. Variation of resilient modulus with soil

suction for compacted subgrade soils. Transportation Research Record. 1913,

Transportation Research Board, Washington, D.C., pp. 99–106.

Yang, S.R., Lin, H.D., Kung, J.H.S. and Huang, W.H. 2008. Suction-controlled

laboratory test on resilient modulus of unsaturated compacted subgrade soils.

Journal of Geotechnical and Geoenvironmental Engineering, 134 (9): 1375–1384.

Yilmaz, I. 2009. Swell potential and shear strength estimation of clays. Applied Clay

Science, 46: 376–384.

Yoshida, R.T., Fredlund, D.G. and Hamilton, J.J. 1983. The prediction of total heave of a

slab-on-grade floor on Regina clay. Canadian Geotechnical Journal, 20(1): 69–81.

Zein, A.K. 1987. Comparsion of measured and predicted swelling behaviour of a

compacted Black Cotton Soil. In Proceedings of the Sixth International Conference

on Expansive Soils. New Delhi, India, pp. 121–126.

Zhan, L.T. 2003. Field and laboratory study of an unsaturated expansive soil associated

with rain-induced slope instability. Ph.D. Thesis. The Hong Kong University of

239

Science and Technology, Hong Kong.

Zhan, L.T., Ng, C.W.W. and Fredlund, D.G. 2006. Instrumentation of an unsaturated

expansive soil slope. Geotechnical Testing Journal, 30(2): 1–11.

Zhan, L.T., Ng, C.W.W. and Fredlund, D.G. 2007. Field study of rainfall infiltration into

grassed unsaturated expansive soil slope. Canadian Geotechnical Journal, 44(4):

392–408.

Zhang, D., Liu, S. and Zhang, T. 2012. Water content and modulus relationship of a

compacted unsaturated soil. Journal of Southeast University (English Edition),

28(2): 209–214.

Zhang, X. 2004. Consolidation theories for saturated-unsaturated soils and numerical

simulations of residential buildings on expansive soils. Ph.D. Thesis. Texas A&M

University, College Station, TX.

Zhang, X. and Briaud, J.L. 2010. Coupled water content method for shrink and swell

predictions. International Journal of Pavement Engineering, 11(1):13–23.

Zhang, X., Lytton, R.L. and Briaud, J.L. 2005. Coupled consolidation theory for

saturated-unsaturated soils. In Proceedings of the Third Biot Conference on

Poromechanics (Biot Centennial), University of Oklahoma, Norman, Oklahoma,

USA, pp. 323–330.

Zumrawi, M. 2013. Swelling potential of compacted expansive soils. International

Journal of Engineering Research and Technology, 2(3).

240